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# Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy

## Abstract

The quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded the concept of topological order in physics bringing into focus the intimate relation between the “bulk” topology and the edge states. The QH effect in graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials. However, the broken-symmetry edge states have eluded spatial measurements. In this article, we spatially map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of $${{\nu }}={{0}},\pm {{1}}$$ across the quantum Hall edge boundary using high-resolution atomic force microscopy (AFM) and show a gapped ground state proceeding from the bulk through to the QH edge boundary. Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moiré superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.

## Introduction

Nontrivial topology is often related to electronic systems with highly degenerate ground states1, the most famous recent example being twisted bilayer graphene displaying the enormous richness of physical phenomena2,3,4. The zero-energy Landau level in graphene is another well-known example of a highly degenerate electronic state. The nontrivial “bulk” topology results in specific topologically protected edge states that have been studied so far with limited success.

The integer QH effect occurs when a two-dimensional (2D) electron system is subjected to a perpendicular magnetic field5,6,7. The metrological precision of the Hall conductance is understood in terms of the topological invariant of the Chern number associated with the Berry connection8,9,10,11. The precise quantization of the Hall conductance is related to the absence of backscattering in topologically protected chiral one-dimensional edge states with opposite momentum directions at the device boundaries. Imaging of QH edge states has been challenging due to their limited spatial extent and their location at the boundaries of the quantum Hall system. A number of notable attempts include: scanning gate microscopy12, scanning single-electron transistor (SET) measurements13, scanning force microscopy14,15, scanning charge accumulation16, and scanning microwave impedance microscopy17. More recent intriguing progress in imaging quantum Hall edge states has been made using SQUID-on-tip measurements of graphene18, however, the authors were not successful in imaging any broken-symmetry states inside the graphene zLL as the technique is limited to a moderate magnetic field range.

The graphene Landau level structure is determined by a combination of the Dirac-like linear energy-momentum dispersion and the π-Berry phase associated with the Dirac point resulting in Landau energies $${E}_{N}=\pm\sqrt{2e{{\hslash }}{v}_{F}^{2}{B|N|}}$$, where e is the elementary charge, $${{\hslash }}$$ is Planck’s constant divided by 2π, $${v}_{F}$$ is the Fermi velocity, B is the magnetic field, and $$N=0,\pm1,\ldots$$ is the Landau level index, resulting in a non-uniform Landau level spacing19,20. The zLL in graphene, with orbital index $$N=0$$, comprises a set of fourfold degenerate Landau levels that are fixed at the Dirac point in the absence of SU(4) symmetry-breaking effects. The nature of the zLL state has been one of intense interest within the framework of quantum Hall ferromagnetism with many competing ground states21,22,23,24,25,26,27. The richness of physics in this regime is dominated by the interplay between the Zeeman energy against the sublattice anisotropy of Coulomb interactions which lift the degeneracy of the SU(4) multiplet. Indeed, extensive theoretical studies of the ground state of the zLL have shown the existence of many competing phases with distinct symmetry-breaking properties21,22,23,24,25,26,27. Among these phases are the ferromagnet (F) state and the antiferromagnet (AF) state, where the latter may form a canted antiferromagnetic (CAF) state24. Other possible phases include a charge density wave (CDW) and a Kekulé state (KD) (see Fig. 18 in ref. 24 for a phase diagram of the zLL). The delicate balance between various competing interactions can be impacted by multiple factors. For example, transitions between different ground states may be induced by changing the contribution of the Zeeman energy by tilting the magnetic field with respect to the graphene sheet26, and through other microscopic variables which break sublattice symmetry, the prominent example being the moiré-induced superlattice27.

The various ground states of zLL are predicted to have qualitatively different edge dispersions and excitations, which might or might not be revealed in a conventional transport experiment. For example, near the quantum Hall sample boundary, the isospin states may disperse into positively dispersing (electron-like) and negatively dispersing (hole-like) states, leading to gapped or gapless edge modes, depending on specific ground-state symmetries21,26,28,29. The edge dispersion of the CAF ground state was predicted to change from gapped to gapless as a function of tilted magnetic field28. Recent experimental observations of a metal–insulator transition observed in transport measurements as a function of tilted magnetic field26 have been interpreted as the evidence of the CAF ground state in accordance with theoretical predictions28. More recent theory29, however, has shown that the metal–insulator transition observed in transport measurements cannot be used as an unambiguous identifier of the zLL ground state in graphene. More complex behavior is expected as the ground-state order parameter changes in the proximity of the quantum Hall boundary29. Imaging the spatial properties of these broken-symmetry states can thus shed light on revealing the competing interactions and make a direct connection with theoretical models.

In this article, we determine the energies and the spatial dispersion of the zLL in graphene with AFM Kelvin probe measurements at arbitrary filling factors. The degeneracy of the zLL observed in AFM measurements is lifted in a magnetic field with the zLL split into four sublevels. The energy splitting of the four sublevels is much larger than the Zeeman energy, indicating interaction-dominated physics. The spatial dispersions of the states at $$\nu =0,\pm 1$$ are measured demonstrating completely gapped spectra as the states progress from the bulk to edge boundary. We discuss these findings in terms of recent theoretical developments of the graphene ground-state properties.

## Results

To image the QH edge states for this study, we chose to use a dual-gated graphene device where the Hall bar boundary is defined by a lateral junction controlled by two independent back gates (Fig. 1)30,31,32. The major advantage of this approach is the atom-scale cleanliness and precision of the boundary that is free of defects and contamination with the graphene lattice perfectly continuous across quantum Hall edge interface, as opposed to a physical boundary of a graphene sheet-shaped using harsh treatments, such as reactive ion etching. A similar device was used in imaging the edge states by SQUID-on-tip measurements of graphene18. Moreover, the area on one “external” side of the boundary can be tuned to an “electronic” insulator by setting the carrier density to zero (filling factor $$\nu =0$$) corresponding to an insulating state of graphene at high magnetic fields33. Such definition of boundaries of graphene devices has led to superior quantum Hall signatures in various geometries34,35,36.

Figure 1a shows the schematic of the layered structure of the device and Fig. 1b is an optical image of the dual-gated device. The overlapped area determined by the graphene boundary (black dotted line) and local gate G1 boundary (red dashed line) in Fig. 1b defines the interior carrier density (area enclosed by the red dashed trapezoid in Fig. 2a). The second graphite gate G2 defines the exterior density. In most of the following measurements, the outside density is set to zero creating an insulating state at high magnetic fields, hence, defining the Hall bar geometry. The potential profile defined by these two back gates is quite sharp due to the thinness of the hBN gate insulator layers. The potential step is ≈70 nm in width as seen from the chemical potential measurements across the boundary in Fig. 1c. This characteristic scale is a few magnetic lengths at 5 T, providing sufficiently strong confinement for the quantum Hall edge channels.

The alignment of the graphene sheet with the underlying hBN dielectric is an important fabrication parameter often impacting the physical properties. In the case of this device, the graphene sheet is rotated relative to the hBN by about 3.1°, as determined from the moiré superlattice observed in STM topography measurements (Fig. 1d). This superlattice gives rise to an additional sublattice symmetry breaking potential generating a zero-field gap $${\triangle }_{\text{AB}}$$. This potential can further affect the possible ground-state phase diagram adding a competing partially sublattice polarized (PSP) state in addition to the CDW and CAF states27. Recent transport measurements have indicated possible isospin phase transitions between these states as a function of magnetic field27.

Measurements of macroscopic and microscopic quantum Hall properties were made using an instrument that is capable of simultaneous magnetotransport, scanning tunneling microscopy, and AFM measurements on a given device at ultralow temperatures37,38. The instrument was operated at 10 mK for all measurements with a perpendicular magnetic field up to 15 T. Magnetotransport measurements of the Hall and longitudinal resistances are shown in Fig. 2b, c. Broken-symmetry states inside the zLL are seen at filling factors $$\nu =0,\pm1$$, marked by quantized plateaus in the Hall resistance (RXY). The same broken-symmetry states are observed in microscopic AFM measurements detected as changes in the frequency shift (averaged tip-sample force gradient) (Fig. 2e) and in the sensor oscillation amplitude (energy dissipation) (Fig. 2f) seen over a narrow density range around the integer filling factors, $$\nu =0,\pm1,\pm2,\pm3,$$ and $$\pm4$$. The AFM response to the symmetry-breaking states derives from the gapped nature of these states and their associated electronic incompressibility. The formation of an incompressible area under the tip apex leads to changes in the system capacitance and resistance, which alters the electrostatic average force gradient between tip and sample. This varies the sensor resonance frequency and hence enables the detection of the broken-symmetry states (see “Methods”). The contrast of the broken-symmetry states scales with the strength of the electrostatic field between the tip and the sample. It disappears at zero electrostatic field, at the sample bias which balances the work function difference between the probe tip and the sample, which is shown by the solid white line in Fig. 2e. The Landau levels are observed on the white line as plateaus with jumps or oscillations at the transitions between Landau levels at integer filling factors. The behavior will be investigated in more detail in chemical potential measurements using Kelvin probe force microscopy (KPFM) measurements, shown below.

### Kelvin probe force spectroscopy of graphene Landau levels

The frequency shift of the AFM qPlus probe, proportional to the average force gradient, shows an inverted parabolic profile as a function of applied electrostatic potential, characteristic of the electrostatic forces (Supplementary Fig. 3d). The vertex of the parabolic response occurs when the applied potential compensates the contact potential difference (CPD) between the probe and graphene and allows for measurements of the local chemical potential by KPFM39,40. Figure 3a displays KPFM measurements of the CPD as a function of the sample bias vs. back-gate potential G2 for different magnetic fields between 9 T and 15 T. These measurements were made outside the local gated area with the density of both areas kept the same by ramping G1 and G2 together with the appropriate scaling of gate voltages. A series of plateaus and transitions are observed at various back-gate potentials, depending on the magnetic field. Each plateau corresponds to the filling of a particular Landau level, whereas the transitions occur at the incompressible states when the Fermi level is being swept in the gaps between the Landau levels. The data in Fig. 3a show the characteristic graphene Landau level energy structure discussed above, as seen by scaling the sample bias by $$\sqrt{B}$$ and the gate potential by B, as shown in Fig. 3b. The correspondence to the graphene Landau level density of states is indicated by the lineup of the $$N=0,\pm1,\pm2$$ Landau levels in Fig. 3c with the plateaus in Fig. 3b.

On close examination of Fig. 3b, the $$N=0$$ Landau level plateau consists of four distinct smaller plateaus, as shown in the blow-up in Fig. 3e for $$B=15\,{\rm{T}}$$. The four plateaus, with the chemical potential labeled $${\varepsilon }_{i}$$, indicate the complete lifting of the degeneracy of the $$N=0$$ Landau level. A large up and down, “N”-shaped excursion in the chemical potential is observed at the integer filling factors as transitions between the plateaus (see Fig. 3d and “Methods” section). Note that the excursion is characterized by very sharp upward jumps of the chemical potential over a small change of filling factor as the incompressible state is entered. The large excursion, on the order of ≈50 meV for $$B=15\,{\rm{T}}$$, is suggestive of interactions playing a strong role. Indeed, in previous measurements, an oscillating behavior of the chemical potential at integer filling factors was interpreted as exchange enhancement of the single-particle or broken-symmetry gaps due to Pauli exclusion41. Nevertheless, the sign of the excursion in our measurements, the peak followed by the dip, is opposite to that in previous measurements on the zeroth Landau level in graphene42, which requires further theoretical investigation. However, peak-dip patterns with negative compressibility have been seen in the phase transitions between correlated states in recent measurements of the flat-band system twisted bilayer graphene43.

Traditionally, the energies of the broken-symmetry states can be investigated by transport measurements only at integer filling factors assuming an activated behavior. These energies vary greatly between different devices, pointing to disorders contributing to the mobility gaps extracted from such activation measurements. In contrast to transport measurements, KPFM measures directly the local chemical potential (Fig. 3e) over a wide range of filling factors, both when the Fermi level is in the compressible state (plateau) and when it is in the incompressible state (between plateaus), complementing and expanding existing methods. These and other recent experiments44 will stimulate further theoretical analysis of partially filled Landau levels that has been lacking, partly because of a deficit of experimental data.

At integer filling factors, the bare symmetry-breaking potential is usually strongly enhanced by exchange and other correlations. The energy differences between the chemical potential plateaus, as indicated by the arrows in Fig. 3e at 15 T and shown in Fig. 3f for different magnetic fields, do reflect the strength of the lattice-scale symmetry-breaking potential but the degree of the enhancement is likely smaller than that at integer filling factors. The enhancement is still significant, as one sees that the energies are much larger than the Zeeman energy (solid black line), and the largest energy gap across $$\nu =0$$ $$({\varepsilon }_{3}-{\varepsilon }_{2})$$ reaches a value of ≈8 meV at 15 T. The energy gaps show distinct low- and high-field dependencies on the magnetic field. Starting with a plateau or even a slight decrease at fields below 8 T, the $$\nu =0$$ gap scales with $$\sqrt{B}$$ at high fields above 8 T. The scaling as $$\sqrt{B}$$ is consistent with electron interactions playing a dominant role. The $$\nu =\pm1$$ energy gaps, $$({\varepsilon }_{2}-{\varepsilon }_{1})$$ and $$({\varepsilon }_{4}-{\varepsilon }_{3})$$, show lower energies monotonically increasing to the highest field of 15 T. As noted above, the measured values of the $$\nu =\pm1$$ energy gaps are larger than the Zeeman energy. Interestingly, the $$\nu =0$$ and $$\nu =\pm1$$ energy gap dependencies can be interpreted as an avoided crossing at around 6–8 T, suggesting a possible change in the ground state. An isospin phase transition proposed in Ref. 27 occurs due to the influence of a moiré superlattice contribution. Indeed, the atomic resolution STM measurements of the device in Fig. 1a does show a moiré period of ≈4.36 nm corresponding to a misalignment of 3.1° of the graphene lattice relative to the hBN underlayer (Fig. 1d). An additional potential of the moiré superlattice causing the sublattice-symmetry breaking is generating a zero-field gap $${\triangle }_{\text{AB}}$$. This alters the possible ground-state phase diagram to include a partial sublattice polarized (PSP) state in addition to the CDW and CAF states27. Magnetotransport measurements as a function of misalignment angle have shown that a zero-field gap scales with rotation angle45, and a gap value of 5–10 meV can be expected for the angle of 3.1°. This value is consistent with the behavior of $$\triangle E(\nu =0)$$ in Fig. 3f if the trend of the low-field regime is extrapolated to zero field (blue line). Respectively, the apparent avoided crossing seen in Fig. 3f at fields of 6–8 T could also be suggestive of a possible isospin transition between a CDW to AF phase at intermediate magnetic fields.

### The quantum Hall edge wedding cake-like potential profile

The edge states in the QH effect form a series of incompressible and compressible strips near the boundary edge (Fig. 4a). The strips originate from the Landau levels that are bent by the potential rise at the boundary and are pinned at the Fermi level (Fig. 4b). In a noninteracting picture, the levels intersect the Fermi-level forming one-dimensional edge states. In reality, interactions and screening reconstruct the potential into a series of plateaus forming a “wedding cake-like” structure (Fig. 4c)46,47. A compressible strip is formed when a partially filled Landau level is at the Fermi level, which is separated by incompressible strips as the next Landau level transitions to the Fermi level. The stepped potential profile near the boundary has been predicted theoretically for many years but has eluded measurement46. Using AFM Kelvin probe spectroscopy, a spatial visualization of the Landau levels from $$N=-2$$ [LL(-2)] to $$N=+1$$ [LL(+1)] is obtained as a function of Y-position across the QH edge boundary (indicated in the black circle in Fig. 2a) and local gate voltage G1, as shown in the Kelvin probe map at $$B=5\,\text{T}$$ in Fig. 4d. Figure 4e shows the extracted stepped potential across the QH edge boundary (see white arrows in Fig. 4d), transformed by screening from the bare potential in Fig. 1c. The profiles in Fig. 4e show the “wedding cake-like” steps with plateaus separated by sharp drops at the incompressible states, as theoretically predicted46. For the larger gate voltage (red line), we observe three incompressible strips corresponding to filling factor $$\nu =-6,-2,{\rm{and}}-1$$. The width of the $$\nu =-6\,{\rm{and}}\,\nu =-2$$, which occurs at the change of the Landau levels are on the order of ≈40 nm. The width of the $$\nu =-1$$ strip is much narrower, on the order of ≈20 nm. These are consistent with the length scale of the electrostatic potential of the graphite back gates defining the quantum Hall edge shown in Fig. 1c, as calculated below in more detail. A similar stepped profile of graphene Landau level energies has been recently measured in graphene quantum dots by tunneling spectroscopy47.

The staircase shape of the chemical potential near the QH boundary is caused by the screening dependence on the filling factor flattening out the potential in compressible states and displaying potential steps over incompressible strips46,47. The width of the incompressible strips, a, can be estimated using Eq. S25 in ref. 47,

$$a={\left(\frac{4(4\pi {\epsilon }_{0}){\epsilon }_{r}{\triangle E}_{\text{LL}}}{{\pi }^{2}{e}^{2}\frac{\partial n}{\partial y}}\right)}^{\frac{1}{2}}$$
(1)

Here $${\epsilon }_{0}$$ is the vacuum permittivity, $${\epsilon }_{r}$$ is the relative permittivity, e is the elementary charge, $${\triangle E}_{\text{LL}}$$ is the energy gap between Landau levels, and $$\partial n/\partial y$$ is the density gradient at the strip position. For the density gradient, we can use the electric field profile across the boundary measured at zero field (Fig. 1c). Here, the boundary width is observed to be on the order of 70 nm, with a potential gradient of ≈1.7 meV/nm across the boundary. For the Dirac dispersion, the corresponding density gradient is $$\approx {10}^{23}{\text{m}}^{-3}$$ using $${v}_{\text{F}}=1.0\,\times {10}^{6}$$ m/s, $${\epsilon }_{r}=5$$. Substituting the density gradient and 80 meV for energy gap in Eq. (1) yields an incompressible strip width of 34 nm, in good agreement with the measured strips in Fig. 4.

### Spatial mapping of the broken-symmetry edge channels

The essential advantage of KPFM is that the measurement is compensating the CPD between tip and sample and thus minimizes gating effects in the graphene 2DEG. Figure 5 shows the KPFM measurements at $$B=10\,\text{T}$$ of the chemical potential across the QH edge boundary as a function of Y-position and local gate potential, G1. At a distance of about 300 nm from the boundary, the different chemical potential plateaus at the left edge of Fig. 5a correspond to the $$N=0,\pm1,\pm2$$ Landau levels, as in Fig. 3b. As the probe approaches the QH edge boundary, the electron (hole) states disperse to positive (negative) densities. The large excursions seen at integer filling factors in the chemical potential in Fig. 3e are useful as fingerprints for spatial mapping of the incompressible states and edge channels in the zeroth Landau level. Inside the $$N=0$$ Landau level, the $$\nu =\pm1$$ incompressible states and the $${\varepsilon }_{i}$$ compressible channels are observed to follow a dispersion similar to the higher Landau levels, as seen in the higher-resolution measurement in Fig. 5b. Following the spatial dispersion to the boundary, we do not observe a crossing or oscillations in the energy of the edge states predicted for some ground-state phases of the zLL. Spatial mapping of the edge states in the XY plane shows the channels are uniform along the boundary edge, i.e., the X direction (Supplementary Fig. 4). The broken-symmetry states remain gapped starting in the bulk and proceeding to the boundary. The $$\nu =-1$$ incompressible edge channel is seen dispersing to below $$\nu =0$$ and $$\nu =+1$$ disperses above. For comparison, the edge dispersion and Kelvin probe simulated maps expected for the KD, CDW, AF, CAF, and F phases are shown in Fig. 6b–f, respectively28,29. We further compare the experimental dispersion with the predictions of these models in the “Discussion” section below.

## Discussion

Magnetotransport experiments have provided convincing evidence for the interaction-dominated nature of the partially filled zLL in graphene. In high-mobility samples, the observed large longitudinal resistance at $$\nu =0$$ indicates that both bulk and edge states are gapped33. Such states belong to the class of quantum Hall ferromagnets with broken symmetries in the spin-isospin space23. The key physical challenge is identifying how the symmetry is broken as well as understanding all the possible microscopic variables in real systems such as disorder or a moiré superlattice which may alter the balance of anisotropies and change the ground state. To this end, studies that combine microscopic and macroscopic measurements on the same device hold an advantage. Our measurements shown in Fig. 3 confirm the lifting of the fourfold degeneracy of zLL with the energy separation of the $$\nu =0$$ state much larger than $$\nu =\pm1$$, and all of them much larger than the bare Zeeman energy indicating interaction-dominated physics.

Theory and transport experiments have mainly converged that the CAF state is the ground state of the zLL in graphene at $$\nu =0$$, although more recent analysis suggests that the graphene ground-state physics is a complex and unsettled issue29,48. The convergence to the CAF state stems from the prediction that the excitation gap closes in a tilted magnetic field28, as shown schematically in Fig. 6e, and supported by transport measurements of a metal–insulator transition as a function of tilt angle of magnetic field26. However, for pure perpendicular magnetic fields, the valley isospin anisotropy is much larger than the Zeeman energy leading to the CAF state approaching the AF ground state (Fig. 6d)26,28,49. In fact, the CAF state bridges smoothly between the AF and F states as the angle between sublattice spin polarizations varies from $$\pi /2$$ to 0, respectively (see Fig. 6d–f)28. In this regard, the measurements in Fig. 6a are consistent with a CAF ground state.

Figure 6b–f shows possible edge spatial dispersions for other bulk phases including the KD and CDW phases. Both the KD and CDW phases show gapped excitations but now with all four levels separated in the bulk and dispersing to the edge (Fig. 6b, c). Both these phases have dispersions consistent with the results in Fig. 6a, in particular the fourfold splitting in the bulk, and show the best agreement with the experiment in Fig. 6a. Evidence for a KD phase has been observed in recent STM LDOS measurements at zero density50. Note that the ground-state phases are likely density-dependent, and even more complex behavior is possible as a bulk ground state can change its order parameter at the boundary as shown in recent calculations29.

In summary, we have obtained spatial measurements of the dispersion of the broken-symmetry edge channels near the quantum Hall boundary. The measurements alone cannot identify the ground-state symmetry in this particular sample but are consistent with edge profiles of various bulk phases as discussed above. Further measurements are required that can shed light on the ground-state properties such as atomically resolved measurements of the QH wavefunction symmetry as a function of distance near the QH boundary. Finally, we point out the unique benefits of KPFM as a new tool for Landau level spectroscopy which complements scanning tunneling spectroscopy. In addition, we demonstrated the combination of macroscopic transport and microscopic scanning probe measurements which advantageously removes uncertainties for a direct comparison of different techniques ensuring that all microscopic variables are key for defining the physics, such as disorder or moiré superlattices, are exactly the same in both measurement modalities.

## Methods

### Graphene device structure and fabrication

Figure 1a, b shows a schematic cross-section of the graphene device heterostructure and an optical top view. Two single-crystal graphite gate electrodes and single-crystal hBN dielectrics are employed for optimal sample quality. The two graphite back gate regions G1 and G2 are outlined in Fig. 1b. G1 defines the carrier density of the local interior area as indicated by the red dashed line in Fig. 1b, while G2 defines the carrier density in the outer region (blue dashed line). A quantum Hall boundary edge can thus be generated at the edge of the local gate G1.

The heterostructure is assembled from top to bottom (starting from the global graphite gate G2 as the top layer) using the van der Waals transfer technique so that the bottom of the graphene flake, as well as the hBN dielectrics, remain free from contamination during the stacking and subsequent fabrication processes. It is then flipped upside down to expose the graphene surface and deposited onto a 285 nm SiO2/ Si++ substrate before vacuum annealing to remove the polymer film underneath the stack. Electrical connections to the graphene sheet and graphite gate electrodes were made by deposition of the Cr/Pd/Au (2/50/50 nm) metal edge contacts.

All but one electrode contacting the graphene are in contact with both the G1 region and the G2 region. The one outside contact is used to ensure that the G2 region is in the $$\nu =0$$ gapped state during electrical transport and scanning probe measurements. A fan-shaped pattern with gold ridges of 65 nm height is connected to the drain electrode to the graphene sheet for navigation purposes (Fig. 1b). After introducing the device sample to the UHV chamber of the scanning probe microscope (SPM) instrument, it is annealed at a temperature of ≈623 K for ≈3 h to obtain the required cleanliness for SPM measurements.

### Navigating to the device with a scanning probe microscope

Navigating to the central device area with scanning probes is always a difficult challenge. For this purpose, a fan-shaped pattern extending to ≈500 µm at its widest region is utilized. Supplementary Fig. 1 outlines the procedure for navigating to the device area. First, the probe tip is aligned onto the fan-shaped area using an optical telescope while the STM module is in the upper ultrahigh vacuum chamber at room temperature (Supplementary Fig. 1a). Using the probe tip reflection, a tip-sample gap of the order of 100 µm is set at room temperature. The module is then transferred and locked into the dilution refrigerator multimode SPM system where it is cooled to a temperature of 10 mK37,38. The landing region is scanned, and a ridge is identified after approaching the fan-shaped runway surface. STM tunneling current or AFM frequency shift feedback can be used for the approach; for this device, AFM feedback was used for approach and navigation to minimize the degradation of the probe tip due to interacting with the surface. Once a ridge on the fan-shaped area is found, an automated algorithm is used to follow a given ridge to the device area (Supplementary Fig. 1b). This algorithm alternates stepping along the ridge direction and quickly scanning the ridge in a “W”-shaped line. After each “W” scan, the XY piezo motor parameters are adjusted to keep the walking direction along the ridge. The successful application of this routine is shown in (Supplementary Fig. 1c), where AFM traces of the “W” line scans are shown. At certain key places, full AFM scans are made to verify marker features in the devices, as shown in (Supplementary Fig. 1c). After successful navigation, the device region is located as verified by AFM scans of the pattern boundary (Supplementary Fig. 1d). Further navigation is then performed to check the device area and locate an area for edge studies, as indicated by the black circle in Fig. 2a.

### Multimode STM, AFM, and transport instrumentation

The study described in this report is the first to use a newly commissioned multimode system with the capabilities of simultaneous AFM, STM, and magnetotransport measurement38. The system utilizes a dilution refrigerator which operates at a base temperature of 10 mK with magnetic fields up to 15 T perpendicular to the sample plane37. Multimode measurements are accomplished by using custom-designed sample and probe tip holders which feature eight electrical contacts for devices and probe sensors. Magnetotransport measurements were performed using a lock-in amplifier at 25 Hz with a 10 nA source current. The qPlus AFM sensor was a new design that incorporated an integrated excitation electrode on the sensor38. The qPlus sensor had a quality factor of $$Q=1.3\times {10}^{5}$$ and a resonance frequency of f0 = 23.4 kHz at zero magnetic field. For the qPlus sensor, four contacts were wired, two to read out the AFM sensor, one for the STM tunneling current, and one for the sensor excitation. All eight electrical contacts of the device were utilized. For AFM and KPFM measurements, we used the frequency modulation mode with an oscillation amplitude of 2–5 nm. For KPFM, we used 1 Hz to 5 Hz modulation on the sample bias voltage.

### AFM frequency shift measurement of broken-symmetry states

AFM measurements of the frequency shift and dissipation were both sensitive to the occurrence of the broken-symmetry states, and Landau levels in general, as shown in Fig. 2. The sensitivity originates from the frequency shift caused by the capacitive forces due to unbalanced electrostatic potentials between the tip and the sample:

$$\triangle {f}_{C} \sim -\frac{{d}^{2}C\left({f}_{0}\right)}{d{z}^{2}}{\left({V}_{{\rm{B}}}-{V}_{{\rm{CPD}}}\right)}^{2}$$
(2)

where $$C\left({f}_{0}\right)$$ is the capacitance between the tip and the sample at the sensor resonance frequency $${f}_{0}$$, $${V}_{{\rm{CPD}}}$$ is the contact potential difference and $${V}_{{\rm{B}}}$$ is the sample bias. The typical frequency shift curves are shown in (Supplementary Fig. 3d). One can see, for example, that at $${V}_{{\rm{B}}}-{V}_{{\rm{CPD}}}\cong 0.5$$ V along the horizontal axis in (Supplementary Fig. 3d), the frequency shift is $$\approx -0.15$$ Hz.

The additional positive frequency shift at integer filling factors corresponding to the broken symmetry states seen in (Supplementary Fig. 2c, d) derives from the openings of gaps which change the resistance and, as a result, the capacitance $$C\left({f}_{0}\right)$$ of the graphene system at the tip location.

An additional dissipation develops once the complex capacitance $$C\left({f}_{0}\right)$$ acquires a phase lag resulting from a large local resistance of the incompressible region. As the resistance grows further, the capacitance $$C\left({f}_{0}\right)$$ decreases, correspondingly causing a positive frequency shift contribution, while the phase lag, and respectively the additional dissipation vanishes. The experimentally observed positive frequency spikes at the integers can be as large as 10–15% of the total frequency shift from the Coulomb attraction (≈0.02 Hz/0.15 Hz).

In the simplified analysis above, all the changes in $$\triangle {f}_{C}$$ were assigned to the changes in $$C\left({f}_{0}\right)$$ assuming $${V}_{{\rm{CPD}}}$$ and $${V}_{{\rm{B}}}$$ being constant. (Supplementary Fig. 2a) illustrates that over a larger parameter range, the latter two variables contribute most significantly. The frequency shift at the broken-symmetry gaps is a small signal on top of a large background due to the larger frequency shifts caused by changes in $${V}_{{\rm{CPD}}}$$ originating from the normal cyclotron gaps at filling factors of $$\nu =\pm2$$ (along the gate voltage axis), as well as by $${V}_{{\rm{B}}}$$ (the sample bias axis), both determining the total electrostatic force contributions. We followed refs. 51,52 and subtracted a smoothly varying background (black curve) from each frequency shift curve (red and blue curves) and plot the residuals, as shown in (Supplementary Fig. 2b, c). The smooth background averages the original data using a Gaussian filter with a sigma of 0.2 V. Each residual curve is then built into a new frequency shift map, as shown in (Supplementary Fig. 2d) and Fig. 2e.

As mentioned above, the frequency shift data contains contributions from electrostatic forces which give rise to a downward parabolic dependence on the applied sample bias (Supplementary Fig. 3d). The vertex of the parabolic response occurs when the applied potential compensates the contact potential difference (CPD) between the probe and graphene, as illustrated in (Supplementary Fig. 3a–c). By measuring changes in the CPD, we can obtain a measure of the local chemical potential, which responds to changes in Fermi-level position with gate bias, as shown in (Supplementary Fig. 3e). The measurements of the chemical potential in (Supplementary Fig. 3e) were obtained by fitting the parabolic dependence on the sample bias over a window of 300 mV about the vertex. A higher-precision measurement with reduced tip-gating effects is obtained by modulating the sample bias and using lock-in detection to measure the sample bias values compensating the CPD. The force at the modulation frequency ω is:

$${F}_{\omega }=-\frac{1}{2}\frac{{{\mathrm{d}}C}}{{{\mathrm{d}}z}}{\left({V}_{{\rm{B}}+\text{AC}}-{V}_{{\rm{CPD}}}\right)}^{2} \sim -\frac{{{\mathrm{d}}C}}{{{\mathrm{d}}z}}\left({V}_{\text{B}}-{V}_{\text{CPD}}\right){V}_{\text{AC}}$$
(3)

Correspondingly, $$\triangle {f}_{\omega } \sim \frac{{\mathrm{d}}{F}_{\omega }}{{{\mathrm{d}}z}}$$ is nullified when $${V}_{{\rm{B}}}-{V}_{{\rm{CPD}}}=0.$$ This method was used in the Kelvin probe measurements in Figs. 36.

All AFM measurements were made using a constant amplitude signal where the dissipation is measured in the excitation drive signal required to keep the amplitude constant with the exception of the data in Fig. 2e, f (and Supplementary Fig. 2) where a constant excitation signal was used and the dissipation was measured in the amplitude signal channel.

### KPFM chemical potential excursions in the N = 0 Landau level

In the N = 0 Landau level, an “N”-shaped excursion in the chemical potential is observed at the integer filling factors between transitions of the mini-plateaus that make up the N = 0 LL. The origin of this excursion is unclear, as it has the opposite sign to previous measurements of the compressibility at the zeroth LL42 using single-electron transistors, and experimental artifacts need to be considered. However, the feature is quite robust and, at present, we cannot identify it as an artifact based on the following arguments: (1) the excursion appears both in the KPFM chemical potential and the AFM frequency shift and dissipation signals, without and with local tip gating, (2) the feature is seen for wide and narrow strips in different spatial locations (see Fig. 5), indicating that local resistive effects are likely not contributing, and (3) the feature is seen near local integer filling factors (see Fig. 5), indicating that non-local resistive/capacitive effects that can be caused by large (bulk) areas of the sample becoming insulating at integer filling factors are not contributing. In addition, the following arguments against a spurious origin of the signal as: (i) the feature is odd with respect to gate voltage, an artifact due to increased resistance for example would be symmetric about the integer fillings, and (ii) the feature was independent of the direction of gate voltage sweeps.

## Data availability

All data are available in the main text or the Supplementary Materials and are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We thank A. MacDonald, A. Young, and B. Feldman for useful discussions. We also thank David Goldhaber-Gordon and Derrick Boone for assistance in device fabrication, and William Cullen for technical assistance. J.S., D.W., and F.G. acknowledge support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology (NIST), Grant No. 70NANB14H209, through the University of Maryland. S.K. acknowledges support under the Office of Naval Research Grant No. N00014-20-1-2352. M.R.S. acknowledges support under the Cooperative Research Agreement between the Georgetown University and NIST, Grant No. 70NANB18H161, through the NIST/Georgetown PREP program, and funding from the Dutch Research Council (NWO) via a Rubicon grant, Grant No. 019.193EN.026. S.T.L. acknowledges support by NIST and grant 70NANB16H170. J.B. and F.J.G. acknowledge support by Deutsche Forschungsgemeinschaft, SFB1277, project A02. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Numbers JP20H00354 and the CREST(JPMJCR15F3), JST. C.R.D. acknowledges support under the Army Research Office Grant No. W911NF-17-1-0323.

## Author information

Authors

### Contributions

S.K., J.S., D.W., F.G., M.R.S., J.A.S., and N.Z. performed the experiments. Y.Z., F.G., and S.T.L. designed and fabricated the graphene device. J.B., S.R.B., F.J.G., and J.A.S. constructed parts of the instrumentation. K.W. and T.T. grew the hBN crystals used in the graphene device. All authors contributed to writing the manuscript.

### Corresponding authors

Correspondence to Cory R. Dean or Joseph A. Stroscio.

## Ethics declarations

### Competing interests

F.J.G. holds patents on the qPlus sensor. The remaining authors declare no competing interests.

Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Kim, S., Schwenk, J., Walkup, D. et al. Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy. Nat Commun 12, 2852 (2021). https://doi.org/10.1038/s41467-021-22886-7

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41467-021-22886-7


text:
# Need to solve the following differential equation

1. Apr 28, 2004

### simick1712

Sorry if this is in the wrong section, I wasn't sure where to post it.

Can anyone help me - I need to solve the following differential equation and find the values of c and B.

> k is given as -98.3146.
> v = 55 when t = 9
> v = 50 when t = 10

k(v^2) + B = m.(dv/dt)

Am I right in thinking that by separation of variables and integration I get

arctan(v/sqrt(B/k))=kt/m + c

??

But then if so, how do I find the values of c and B?

Thanks for any help,

Simon.

2. Apr 28, 2004

### krab

Your solution would be OK, except that k is negative. Note that you are taking the square root of k. The proper way to do this gets you an inverse hyperbolic tangent, not an arctan.

Also, your k on the rhs should be sqrt(Bk)

Last edited: Apr 28, 2004
3. Jul 5, 2004

### eJavier

First, I don't see any C in the equation to solve, so I assume it's the constant of integration.
I got:
$$\frac{\sqrt{B}}{\sqrt{k}}\tanh{\frac{\sqrt{B}\sqrt{k}\cdot t+C \sqrt{B}\sqrt{k}\cdot m}{m}}$$

Last edited: Jul 5, 2004
4. Jul 5, 2004

### eJavier

You could also try the "homogeneous + particular" approach.

5. Jul 5, 2004

### HallsofIvy

Staff Emeritus
No, you can't because this is a non-linear equation. The whole point of linear equations is that you can solve separate parts of the problem, then put them together. With non-linear equations you can't do that.

6. Jul 5, 2004

### eJavier

Yes, you're right. :blush:

format: plain


text:
# Setting up a client for development

Blockchain development often requires to rely on a test Blockchain, an environment that behaves like a Blockchain without the constraints of a real mainnet. These test-oriented environments are called Testnets or Devchains. Working with a devchain enables you to: - work faster (devchains use a fast consensus model) - work cheaper: you don’t have to pay gas for each transaction (devchains run on “monopoly gas” which doesn’t cost anything) - work safer: in the case of a local Devchain your can keep your work private

Depending on your use-case, you might rely on a public Testnet (like Rinkeby or Gorli) or a local devchain. The main differences between them are that Testnets are not private and require to be online while local devchains are private and can be used offline. It is common practice to start the development on a local devchain and then deploy the code on a Testnet to verify and test assumptions.

The below is a list of the main Ethereum clients and how to set them up as devchains.

## Local devchain clients

All the main Ethereum clients can be configured as Testchains. When working in a local environment, the tools we recommend are Geth Clique, Parity PoA and Ganache CLI

Below are the most used Ethereum clients:

### Geth Clique

Using Geth Clique allows you to test your integration with smart contracts directly in Geth with a fast Proof of Authority consensus.

Geth Clique Manual

### Parity

Using Parity in combination with PoA consensus mechanism yields the same benefits as with Geth.

Parity Clique Manual

### Ganache-CLI

Ganache CLI (formerly Test RPC) is a very common and well-documented chain emulator.

Ganache inherits Truffle suite’s benefits: mainly simplifying Smart Contracts testing, debugging and updating. Ganache has a quasi-immediate response time and great feedback cycle. It’s great to deploy contracts and interact with them instantly at no GAS cost.

Ganache official repo

## Debug mode

Debug modes are available whether you are using chain emulators or full-on Ethereum clients.

### 1 - Geth

• Using option debug

### 2 - Parity

• The use of Json RPC’s Trace Module allows to trace transactions.

### 3 - Ganache/Testnet RPC

• Use the --debug option will Output VM opcodes for debugging

## Cloud private testnet

Azure BaaS (Blockchain as a Service) allows you to deploy a testnet with several nodes and have a team work on the development.

## Public testnets

Public testnets function in the same way as mainnets, with two differences: * They are free: transactions are paid with a worthless crypto-currency * They are often more responsive than main nets

The main public testchains are:

#### 1. Rinkeby

Cross-client testnet using PoA consensus mechanism.

Rinkeby Official Site

#### 2. Kovan

Cross-client testnet (working with both Parity and Geth) using PoA consensus mechanism and a particular focus on spam-resilience.

Kovan Official Site

#### 3. Ropsten

Less popular since hacked early 2017, however, Ropsten is still in service.

Ropsten Github Repo

Note: public testnets can be accessed via public nodes such as INFURA

#### 3. Goerli

The most recent testnet:

Goerli Official Site

## Ether Faucets

The ether and tokens used in test networks is for testing purposes only (it can accurately be compared to Monopoly bills). It’s mostly distributed by online “faucets”.

Here’s a list of testnet Ether sources:

Testnet Name Faucet
Rinkeby https://www.rinkeby.io/#faucet
Ropsten https://blog.b9lab.com/when-we-first-built-our-faucet-we-deployed-it-on-the-morden-testnet-70bfbf4e317e
Kovan Kovan requires you to request KETH from another person
Goerli https://goerli-faucet.slock.it/

## Testchains

At Nethereum, we developed a tool to simplify and accelerate Ethereum Blockchain development. It’s called Testchains, installs in minutes and allows you to code fast with your Ethereum client of choice. Testchains can be downloaded at https://github.com/Nethereum/TestChains

Credits to Karl Floersch for the information about the different testnets: https://karl.tech/intro-guide-to-ethereum-testnets/


text:
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Matrices

# Prove by Mathematical Induction that $(A')^n=(A^n)'$,where $n\in N$ for any square matrix A.

Toolbox:
• We use the principle of mathematical induction, where we need to prove P(n) is true for n=1, n=k, n=k+1
Step1:
Let P(n) be $(A')^n=(A^n)'$
Let P(n) be true for all $n\in N$
For n=1
$(A')^1=(A^1)'$
$\Rightarrow (A)'=A'.$
Hence LHS=RHS.
Hence P(n) is true for n=1
Step2:
Let P(n) be true for n=k.
Put n=k
$(A')^k=(A^k)'$
Multiply $A'$ on both the side
$(A')^k.A'=(A^k)'.A'$
$\Rightarrow {A'^k}.A'=(A^k)'.A'$
$\Rightarrow {A'^k}.A'^1=(A^k.A^1)'$
$\Rightarrow (A')^{k+1}=(A^{k+1})'$
Hence P(n) is true for n=k+1.


text:
Equivalent of Lusin's Theorem in Borel setting

Let $X$ be a Polish space, $\mathcal B$ the sigma-algebra of Borel sets. Let $E$ be an aperiodic countable Borel equivalence relation on $X \times X$ (this means that every class of equivalence is countably infinite). A set $C\in \mathcal B$ is called a complete section for $E$, if $\forall x \in X$ $\exists y \in C$ such that $(x, y) \in E$.

$\textit{Question:}$ Let $f : X \to \mathbb R$ be a Borel real-valued function. Is there a complete section $C$ such the restriction of $f$ on the set $C$, $f|_C$, is a continuous function.

A suitable counterexample can be constructed as follows.

On the real line $\mathbb R$ consider the equivalence relation $E=\{(x,y)\in\mathbb R\times \mathbb R:x-y\in\mathbb Q\}$.

Fix a countable base $\{U_n\}_{n\in\omega}$ of the topology on the real line. Take a countable set $X=\{x_n\}_{n\in\omega}\subset\mathbb R$ such that $x_n-x_m\notin \mathbb Q$ for any distinct $n,m\in\omega$.

In the real line $\mathbb R$ consider the $G_\delta$-set $$G:=\mathbb R\setminus\bigcup_{n\in\omega}(\mathbb R\setminus U_n)\cap(x_n+\mathbb Q)=(\mathbb R\setminus(X+\mathbb Q))\cup\bigcup_{n\in\omega}U_n\cap(x_n+\mathbb Q)$$ and the Borel function $f:G\to\mathbb R$ $$f(x)=\begin{cases} n&\mbox{if x\in U_n\cap (x_n+\mathbb Q) for some n\in\omega};\\ 0&\mbox{otherwise}. \end{cases}$$ It is easy to see that the equivalence relation $E_G:=E\cap(G\times G)$ on $G$ has no complete section $C$ with continuous restriction $f|C$; moreover, for any complete section $C$ of $E_G$, the restriction $f|C$ has no continuity points.

There are (a bit more involved) counterexamples even for very good equivalence relations.

To construct a suitable example, take the convergent sequence $S:=\{0\}\cup\{2^{-n}:n\ge0\}\subset\mathbb R$ and on the compact zero-dimensional space $X:=S^\omega\times S$ consider the equivalence relation $$E=\{((x,y),(x',y'))\in X\times X:x=x'\}.$$ Observe that the quotient map $q:X\to X/E=S^\omega$ is open and closed.

Now take any bijective map $p:S\to D$ to a countable space $D$. Its countable power $P:S^\omega\to D^\omega$, $P:(x_n)_{n\in\omega}\mapsto (p(x_n))_{n\in\omega}$, is known as the Pawlikowski function and is a standard example of a Borel function of the first Baire class, which is not $\sigma$-continuous. More precisely, the Pawlikowski function has the property that a subset $C\subset S^\omega$ is nowhere dense in $S^\omega$ if the restriction $P|C$ is continuous.

We claim that the Borel function $f:=P\circ q:X\to D^\omega$ and the equivalence relation $E$ yield a counterexample to the question posed by Shrey. Indeed, assume that the equivalence relation $E$ has a complete section $C$ with continuous restriction $f|C$. For every $s\in S$ let $C_s:=\{x\in S^\omega:(x,s)\in C\}$ and observe that the continuity of the restriction $f|C$ implies the continuity of the map $P|C_s$ and hence nowhere density of $C_s$ in $S^\omega$. Since the space $S^\omega$ is compact and hence Baire, we conclude that $\bigcup_{s\in S}C_s\ne S^\omega$, which contradicts the choice of $C$ as a complete section of $E$.

• Thank you for your reply. Edit: Let me edit this problem to ask the following : Is there a complete section $C$ such the restriction of $f$ on the set $C$, $f|_C$, is bounded. – Shrey Jul 26 '18 at 17:32
• @Shrey It seems that the first example has the unboundedness property you required. – Taras Banakh Jul 26 '18 at 19:52
• Thank you. If you allow me to ask further. would it be possible to do so if I restrict my equivalence class to aperiodic non-smooth hyperfinite Borel equivalence relation. Or is there any other case where it is possible. – Shrey Jul 26 '18 at 20:26
• @Shrey Sorry, this is too complicated for me. I do not know what is hyperfinite and non-smooth equivalence relation. – Taras Banakh Jul 26 '18 at 20:32
• You don't need to be sorry. Thank you for all you help. I really appreciate it. – Shrey Jul 26 '18 at 20:37


text:
## Continuity of eigenvalues

As a simple corollary to the result from my previous post I would like to show you how to obtain continuous dependence of the eigenvalues of a matrix on its entries. Specifically, let be the complex vector space of matrices with entries in , endowed with any norm of our… Read moreContinuity of eigenvalues


text:
# Make my XML Parser in Java using WoodStox run faster and use less memory, or just generally better

I have an XML Parser in Java using WoodStox that I wrote. This parser is going to be parsing through extremely large files, could be 5+GB. The goal of the parser is to convert a nest XML file into a CSV. The XML file is going to be formatted in such a way where there will be a 'rowTag' that has the actual information that the parser is interested in. Take for example XML file:

<persons>
<person id="1">
<firstname>James</firstname>
<lastname>Smith</lastname>
<middlename></middlename>
<dob_year>1980</dob_year>
<dob_month>1</dob_month>
<gender>M</gender>
<salary currency="Euro">10000</salary>
<street>456 apple street</street>
<city>newark</city>
<state>DE</state>
</person>
<person id="2">
<firstname>Michael</firstname>
<lastname></lastname>
<middlename>Rose</middlename>
<dob_year>1990</dob_year>
<dob_month>6</dob_month>
<gender>M</gender>
<salary currency="Dollor">10000</salary>
<street>4367 orange st</street>
<city>sandiago</city>
<state>CA</state>
</person>
</persons>

The rowtag here would be "person", and the headers would be all the tags inside of <person>.

Below is the class I wrote for this. Would love to hear feedback. You can use WoodStox with either this jar file here, or include it in gradle:

implementation(["org.codehaus.woodstox:stax2-api:3.1.1"])

import java.io.BufferedWriter;
import java.io.File;
import java.io.FileWriter;
import java.io.IOException;
import java.nio.file.Files;
import javax.xml.parsers.FactoryConfigurationError;
import javax.xml.stream.XMLInputFactory;
import javax.xml.stream.XMLStreamException;
import org.apache.commons.lang3.StringUtils;
import org.codehaus.stax2.XMLInputFactory2;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;

public class XmlConverter2
{
private static final Logger logger = LoggerFactory.getLogger(XmlConverter.class);
private static final String ROWTAG = "person";

throws FactoryConfigurationError, XMLStreamException, IOException
{
long startTime = System.nanoTime();
// set up a Woodstox reader
XMLInputFactory xmlif = XMLInputFactory2.newInstance();
boolean firstPass = true;
boolean insideRowTag = false;
Files.deleteIfExists(new File( file + ".csv").toPath());
BufferedWriter br = new BufferedWriter(new FileWriter(file + ".csv", true), 64*1024*1024);
StringBuilder firstItems = new StringBuilder();

try
{
{

// If 4 event, meaning just some random '\n' or something, we skip.
{
continue;
}

// If we are at a start element, we want to check a couple of things
{
// If we are at our rowtag, we want to start looking at what is inside.
// We are 'continuing' because a Rowtag will not have any "elementText" in it, so we want to continue to the next tag.
{
insideRowTag = true;
continue;
}

// if we are at a tag inside a row tag, we want to extract that information (the text it contains) from it....
if (insideRowTag)
{
// ...but first, if we have not started to collect everything, we need to collect the headers!
// This makes an assumption that all the "headers" are constant. If the first record has 6 tags in it,
// but the next one has 7 tags in it, we are in trouble. We can add flexibility for that, I think.
if (firstPass)
{
// We want to write the headers first

// And collect the items inside in a stringBuilder, which we'll dump later.
} else
{
// If we're not in the first pass, just write the elements directly.
}
}
}

// If we are at an end element that is the rowTag, so at the end of the record, we want to do a couple of things
{
// First, if we are at the first pass, we want to send out the elements inside the first record
// that we were collecting to dump *after* we got all the headers
if (firstPass)
{
firstPass = false;
br.write('\n' + StringUtils.chop(firstItems.toString()));
}

// Then we set this off so that we no longer collect irrelevant data if it is present.
insideRowTag = false;
br.write('\n');
}

}
}
catch (Exception e)
{
logger.error("Error! " + e.toString());
}
finally
{
}
br.close();

long endTime   = System.nanoTime();
long totalTime = endTime - startTime;
logger.info("Done! Time took: {}", totalTime / 1000000000);
}
}

My goal is to make this faster and/or consume less memory. Any other advice is appreciated, of course. I am executing it using -Xms4g -Xmx4g tags. Right now, it takes around 25 seconds to run on an xml file that is 1.5Gb approximately.

• It's a bit odd to call your program a "parser". Woodstox is the parser. Jan 30, 2020 at 20:33
• I think 1.5Gb in 25s is a pretty good speed, and I wouldn't think it can be improved all that much. Try writing an application that does the absolute minimum to invoke Woodstox to parse the XML, and I suspect it won't be much faster than this. The critical component from a performance perspective is almost certainly the Woodstox parser, and not your calling application. There might be a few tiny savings you can make, for example detecting the ROW end tag by tracking nesting depth rather than with a string comparison; but they're going to be very minor. Jan 30, 2020 at 20:39
• In fact since you aren't handling nested rowtags anyway, you could avoid the string comparison on start tag names when inrowtag is true. Jan 30, 2020 at 20:44
• @mtj in my experience it matters a lot more if a parser allocates Strings at any point or only juggles some internal buffers, giving you some sort of a view on it. Feb 1, 2020 at 10:36
• FileReader and FileWriter use the default platform encoding; probably wrong. Nice would be UTF-8. For XML better use an InputStream. If so needed with a BOM as first char "\uFEFF". I also have some doubt on skipping isCharacters but I assume that is for the header line. Feb 3, 2020 at 11:37

I have some suggestion for the code, that will not make to code faster, but cleaner, in my opinion.

1. You can use the try-with-resources to handle the closing of the stream automatically (java 8+)
try(BufferedWriter br = new BufferedWriter(new FileWriter(file + ".csv", true), 64 * 1024 * 1024)) {
//[...]
}

1. I suggest that you separate the code in more methods, to separate the different sections; preferably the section that handles the reading / parsing of the XML.
private void parseXml(XMLStreamReader xmlStreamReader, boolean firstPass, boolean insideRowTag, BufferedWriter br) throws XMLStreamException, IOException {
StringBuilder firstItems = new StringBuilder();

// If 4 event, meaning just some random '\n' or something, we skip.
continue;
}

// If we are at a start element, we want to check a couple of things
// If we are at our rowtag, we want to start looking at what is inside.
// We are 'continuing' because a Rowtag will not have any "elementText" in it, so we want to continue to the next tag.
insideRowTag = true;
continue;
}

// if we are at a tag inside a row tag, we want to extract that information (the text it contains) from it....
if (insideRowTag) {
// ...but first, if we have not started to collect everything, we need to collect the headers!
// This makes an assumption that all the "headers" are constant. If the first record has 6 tags in it,
// but the next one has 7 tags in it, we are in trouble. We can add flexibility for that, I think.
if (firstPass) {
// We want to write the headers first

// And collect the items inside in a stringBuilder, which we'll dump later.
} else {
// If we're not in the first pass, just write the elements directly.
}
}
}

// If we are at an end element that is the rowTag, so at the end of the record, we want to do a couple of things
// First, if we are at the first pass, we want to send out the elements inside the first record
// that we were collecting to dump *after* we got all the headers
if (firstPass) {
firstPass = false;
br.write('\n' + StringUtils.chop(firstItems.toString()));
}

// Then we set this off so that we no longer collect irrelevant data if it is present.
insideRowTag = false;
br.write('\n');
}
}
}

### Refactored code

public class XmlConverter2 {
private static final Logger logger = LoggerFactory.getLogger(XmlConverter2.class);
private static final String ROWTAG = "person";

throws FactoryConfigurationError, XMLStreamException, IOException {
long startTime = System.nanoTime();
// set up a Woodstox reader
XMLInputFactory xmlif = XMLInputFactory2.newInstance();

boolean firstPass = true;
boolean insideRowTag = false;

Files.deleteIfExists(new File(file + ".csv").toPath());

try (BufferedWriter br = new BufferedWriter(new FileWriter(file + ".csv", true), 64 * 1024 * 1024)) {
} catch (Exception e) {
logger.error("Error! " + e.toString());
} finally {
}

long endTime = System.nanoTime();
long totalTime = endTime - startTime;
logger.info("Done! Time took: {}", totalTime / 1000000000);
}

private void parseXml(XMLStreamReader xmlStreamReader, boolean firstPass, boolean insideRowTag, BufferedWriter br) throws XMLStreamException, IOException {
StringBuilder firstItems = new StringBuilder();

// If 4 event, meaning just some random '\n' or something, we skip.
continue;
}

// If we are at a start element, we want to check a couple of things
// If we are at our rowtag, we want to start looking at what is inside.
// We are 'continuing' because a Rowtag will not have any "elementText" in it, so we want to continue to the next tag.
insideRowTag = true;
continue;
}

// if we are at a tag inside a row tag, we want to extract that information (the text it contains) from it....
if (insideRowTag) {
// ...but first, if we have not started to collect everything, we need to collect the headers!
// This makes an assumption that all the "headers" are constant. If the first record has 6 tags in it,
// but the next one has 7 tags in it, we are in trouble. We can add flexibility for that, I think.
if (firstPass) {
// We want to write the headers first

// And collect the items inside in a stringBuilder, which we'll dump later.
} else {
// If we're not in the first pass, just write the elements directly.
}
}
}

// If we are at an end element that is the rowTag, so at the end of the record, we want to do a couple of things
// First, if we are at the first pass, we want to send out the elements inside the first record
// that we were collecting to dump *after* we got all the headers
if (firstPass) {
firstPass = false;
br.write('\n' + StringUtils.chop(firstItems.toString()));
}

// Then we set this off so that we no longer collect irrelevant data if it is present.
insideRowTag = false;
br.write('\n');
}
}
}
}

• This definitely looks a lot cleaner! Thank you for the suggestion! Feb 1, 2020 at 10:41

I was somewhat curious if Woodstox has improved, so I wrote a complete parser for your example data. It's in a different style than your code, complete repo: https://github.com/chhh/testing-woodstox-xml-parsing

My results with fake data records that I created:
Parsed 4,000,000 persons (1.36 GB) in 16.75 seconds (Ryzen5 3600), memory usage wasn't really significant.

First of all there's a newer version of Woodstox on Maven Central.
Gradle dependency: implementation 'com.fasterxml.woodstox:woodstox-core:6.0.3'
They now have XMLStreamReader2 with .configureForSpeed() option. I didn't really check what it does, but for my test it didn't do much.

Had to create fake data. You can make files of any size with FakeData.createHugeXml(Path path, int numEntries).

Just in case, here's main parsing code, excluding the Person class (which is not very interesing and can be found here)

public class WoodstoxParser {
@FunctionalInterface
interface ConditionCallback {
}

interface TagPairCallback {
void tagStart(String tagName, XMLStreamReader2 sr) throws XMLStreamException;

void tagContents(String tagName, StringBuilder sb);
}

public static void processUntilTrue(XMLStreamReader2 sr, ConditionCallback callback) throws XMLStreamException {
do {
if (callback.processXml(sr))
return;
} while (sr.hasNext() && sr.next() >= 0);
throw new IllegalStateException("xml document ended without callback returning true");
}

/** Main parsing function. **/
public static List<Person> parse(Path path) throws IOException, XMLStreamException {
XMLInputFactory2 f = (XMLInputFactory2) XMLInputFactory2.newFactory();
f.configureForSpeed();
//        f.configureForLowMemUsage();

// fast forward to beginning 'persons' tag (will throw if we don't find the tag at all)
processUntilTrue(sr, sr1 -> isTagStart(sr1, "persons"));

final List<Person> persons = new ArrayList<>(); // we've found the tag, so we can allocate storage for data
final StringBuilder sb = new StringBuilder(); // reuse a single string builder for all character aggregation

// now keep processing unless we reach closing 'persons' tag
processUntilTrue(sr, sr1 -> {
if (isTagEnd(sr1, "persons"))
return true;

if (isTagStart(sr1, "person")) {
// now we're finally reached a 'person', can start processing it
int idIndex = sr1.getAttributeInfo().findAttributeIndex("", "id");
Person p = new Person(Integer.parseInt(sr1.getAttributeValue(idIndex)));

sr1.next();
processUntilTrue(sr1, sr2 -> {
// processing the meat of a 'person' tag
// split it into a function of its own to not clutter the main loop
//return processPerson(sr2, p, sb);
if (isTagEnd(sr2, "person"))
return true; // we're done processing a 'person' only when we reach the ending 'person' tag

if (isTagStart(sr2))
processTagPair(sr2, sb, p);

return false;
});
// we've reached the end of a 'person'
if (p.isComplete()) {
} else {
throw new IllegalStateException("Whoa, a person had incomplete data");
}
}

return false;
});
return persons;

} finally {
if (sr != null)
sr.close();
}

}

public static void processTagPair(XMLStreamReader2 sr, StringBuilder sb, TagPairCallback callback) throws XMLStreamException {
final String tagName = sr.getLocalName();
callback.tagStart(tagName, sr); // let the caller do whatever they need with the tag name and attributes
sb.setLength(0); // clear our buffer, preparing to read the characters inside
processUntilTrue(sr, sr1 -> {
switch (sr1.getEventType()) {
callback.tagContents(tagName, sb); // let the caller do whatever they need with text contents of the tag
return true;
sb.append(sr1.getText());
break;
}
return false;
});
}

public static boolean isTagStart(XMLStreamReader2 sr, String tagName) {
return XMLStreamReader2.START_ELEMENT == sr.getEventType() && tagName.equalsIgnoreCase(sr.getLocalName());
}

public static boolean isTagStart(XMLStreamReader2 sr) {
}

public static boolean isTagEnd(XMLStreamReader2 sr, String tagName) {
return XMLStreamReader2.END_ELEMENT == sr.getEventType() && tagName.equalsIgnoreCase(sr.getLocalName());
}
}

• Thank you so much for the answer. I’m currently on vacation and get back in two or so weeks. Will take a closer look then - unless I get curios, which I probably will, and look at it before :) Feb 1, 2020 at 10:40
• According to xml.com/pub/a/2007/05/09/xml-parser-benchmarks-part-1.html (which is quite old), Javolotion parsing performance is pretty much the same as Woodstox - fast, but not wildly faster than other parsers. Jan 31, 2020 at 9:50
• @MichaelKay Just sharing my experience. I wasn't able to squeeze out comparable performance out of woodstox, but maybe I did domething wrong somewhere. Jan 31, 2020 at 19:27

Aside from other suggestions which make sense, one simple thing to try is to use Aalto-xml parser from:

https://github.com/FasterXML/aalto-xml

it implements Stax API (as well as Stax2 extension, SAX); and as long as you do not need full DTD handling (which I suspect you don't) has the feature set you need. For common read use cases I think it can be 30-40% faster; but most importantly it should be very easy to just try out.

Maven coordinates are:

• group id: com.fasterxml
• artifact id: aalto-xml
• version (latest): 1.2.2

And XMLInputFactory implementation com.fasterxml.aalto.stax.InputFactoryImpl. I would recommend creating instance directly, instead of using XMLInputFactory.newInstance() so you can sure of the exact implementation you have (if you have multiple Stax implementations in classpath, choice is arbitrary).


text:
# Supremum or infimum of a function

Does there exist an elementary function on some subset $I$ of $R$ such that you can prove that $A = \{\sup(f(x)): x \in I\}$ exists, but you cannot find the value of $A$ ?

If the answer is "no", then replace "elementary function" above by "function" as a further question

-
Does something like $A = \sup \{1 - \frac 1x \mid x \in 2\mathbb Z, x \ge 4, \exists p,q \text{ prime }: x = p+q\}$ count for you? – martini Nov 6 '12 at 14:23
Ha. Good points. Will have a think about that and get back to you later. – Adam Rubinson Nov 6 '12 at 14:30
Martini - your comment only makes sense if you are able to prove that A exists. Which you are not. Will - what is meant by "cannot find" is self-evident. And I don't see any problems with doing this. – Adam Rubinson Nov 7 '12 at 13:40
Bumpety bump bump – Adam Rubinson Nov 12 '12 at 13:56
Ah..... A exists because we do know that {x: x is even, x = p+q where p and q are prime} is unbounded. – Adam Rubinson Nov 13 '12 at 13:54


text:
QUESTION

BUS 201 Week 2 DQ 1 Environmental Factors In Globalization

In this pack of BUS 201 Week 2 DQ 1 Environmental Factors In Globalization you will find the next information:

Globalization is the set of specific and general forces that work together to integrate and connect economic, political and social systems across countries, cultures, or geographical regions. The result of globalization is that nations and peoples become increasingly interdependent because the world

• @
• 1 order completed
Tutor has posted answer for $5.19. See answer's preview$5.19


text:
# Methanol reformer

A methanol reformer is a device used in chemical engineering, especially in the area of fuel cell technology, which can produce pure hydrogen gas and carbon dioxide by reacting a methanol and water (steam) mixture.

${\displaystyle \mathrm {CH_{3}OH_{(g)}+H_{2}O_{(g)}\;\longrightarrow \;CO_{2}+3\ H_{2}\qquad } \Delta H_{R\ 298}^{0}=49.2\ \mathrm {kJ/mol} }$

Methanol is transformed into hydrogen and carbon dioxide by pressure and heat and interaction with a catalyst.

## Technology

A mixture of water and methanol with a molar concentration ratio (water:methanol) of 1.0 - 1.5 is pressurized to approximately 20 bar, vaporized and heated to a temperature of 250 - 360 °C. The hydrogen that is created is separated through the use of Pressure swing adsorption or a hydrogen-permeable membrane made of polymer or a palladium alloy.

There are two basic methods of conducting this process.

• The water-methanol mixture is introduced into a tube-shaped reactor where it makes contact with the catalyst. Hydrogen is then separated from the other reactants and products in a later chamber, either by pressure swing adsorption (PSA), or through use of a membrane where the majority of the hydrogen passes through. This method is typically used for larger, non-mobile units.
• The other process features an integrated reaction chamber and separation membrane, a membrane reactor. In this, relatively new approach, the reaction chamber is made to contain high-temperature, hydrogen-permeable membranes that can be formed of refractory metals, palladium alloys, or a PdAg-coated ceramic. The hydrogen is thereby separated out of the reaction chamber as the reaction proceeds, This purifies the hydrogen and, as the reaction continues, increases both the reaction rate and the amount of hyrogen extracted.

With either design, not all of the hydrogen is removed from the product gases (raffinate). Since the remaining gas mixture still contains a significant amount of chemical energy, it is often mixed with air and burned to provide heat for the endothermic reforming reaction.

Methanol reformers are being considered as a component of a hydrogen fuel cell-powered vehicle. A prototype car, the NECAR 5, was introduced by Daimler-Chrysler in the year 2000. The primary advantage of a vehicle with a reformer is that it does not need a pressurized gas tank to store hydrogen fuel; instead methanol is stored as a liquid. The logistic implications of this are great; pressurized hydrogen is difficult to store and produce. Also, this could help ease the public's concern over the danger of hydrogen and thereby make fuel cell powered vehicles more attractive. However, methanol, like gasoline, is toxic and (of course) flammable. The cost of the PdAg membrane and its susceptibility to damage by temperature changes provide obstacles to adoption.

While hydrogen power produces energy without CO2, a methanol reformer creates the gas as a byproduct.

Methanol (prepared from natural gas) that is used in an efficient fuel cell, however, releases less CO2 in the atmosphere than gasoline, in a net analysis.[1]

## References

1. ^ George A. Olah (2005). "Beyond Oil and Gas: The Methanol Economy". Angewandte Chemie International Edition 44 (18): 2636–2639. doi:10.1002/anie.200462121
• Emonts, B. et al.: Compact methanol reformer test for fuel-cell powered light-duty vehicles, J. Power Sources 71 (1998) 288-293
• Wiese, W. et al.: Methanol steam reforming in a fuel cell drive system, J. Power Sources 84 (1999) 187-193
• Peters, R. et al.: Investigation of a methanol concept considering the particular impact of dynamics and long-term stability for use in a fuel-cell-powered passenger car, J. Power Sources 86 (1999) 507-514


text:
# 7.3. Background species¶

## 7.3.1. Introduction¶

Background species have first been introduced in Osmose by Fu Caihong (see [FOT+17]). They can be viewed as an intermediary between focal species (i.e. species of interest, whose full life cycle is simulated) and lower trophic levels (plankton for instance). They differ from lower trophic levels since they can predate on and be predated by focal species.

Background species are activated by setting the parameter simulation.nbackground to values greater than 0.

## 7.3.2. BackgroundSpecies class¶

To manage background species, a new class has been implemented (BackgroundSpecies.java). It contains the following attributes:

• index of the background species (used in the configuration files)

• ByClassTimeSeries object, which allows to extract biomass time series by time-step and by size class. It is defined by the "biomass.byDt.bySize.file.bkgX parameter.

Warning

For background species, the size classes are fixed!

• The name of the background species (species.name.bkgX parameter)

• Allometric factors for size/weight conversions (species.length2weight.condition.factor.bkgX and species.length2weight.allometric.power.bkgX parameters).

• Distribution maps. They can be provided as CSV files, in the same way as focal species movements (see Section 4.8), with parameters starting with movement.bkgspecies. They can also be defined as NetCDF files (one per background species), which should have the following format:

netcdf movement_bkg0 {
dimensions:
m = 3 ;
y = 22 ;
x = 45 ;
t = 24 ;
variables:
int yearmax(m) ;
double map(m, y, x) ;
map:_FillValue = NaN ;
int class(m) ;
int yearmin(m) ;
byte season(m, t) ;
}

• Trophic levels for each size class (species.trophiclevel.bkgX parameters).

In addition, the background species need to have the predation parameters that are also used by focal species ( predation.accessibility.stage.threshold.bkgX predation.efficiency.critical.bkgX predation.ingestion.rate.max.bkgX predation.predPrey.stage.threshold.bkgX predation.predPrey.sizeRatio.max.bkgX predation.predPrey.sizeRatio.min.bkgX, see Section 4.10)

## 7.3.3. Accessibility matrix¶

When using background species, the accessibility matrix must be changed accordingly. It must always have the following form:

 Focal (pred) Background (pred) Focal (prey) Background (prey) LTL (prey)

## 7.3.4. Background biomass¶

Background species biomass is defined from a biomass time series (one per species and per size class) and by distribution maps. The distribution maps contain defines the distribution of the background species over space. They contain float values, which are normalize, so that the integral over space equals one:

$\sum_{k=0}^{N_{cell}-1} D_k = 1$

Before each time step, the background species biomass is reset by multiplying the biomass time series by the map distribution factor. Because of the normalisation, the spatially integrated biomass is equal to the biomass provided in the time-series.

## 7.3.5. Mortality¶

Background species come into play in the predation mortality process.

Warning

Background species have only be implemented in the stochastic mortality algorithm (StochasticMortalityProcess.java).

For each cell:

• The list of background species schools is recovered. As for LTL, it is assumed that each cell contain one unique school for each background species and for each size class. This list is added to the list of potential preys (i.e. school of focal species, swarms of ltl that belong to the cell).

• For all the background schools, the accessibility to all the preys is estimated.

• The biomass of all the schools is initialized (see Section 7.3.4).

• A random draft of predators (both focal and background species) is performed. And for each predator, the predation mortality toward each prey (focal, background and ltl species) is computed.

• The number of dead individuals are then removed.


text:
694 miles per hour in meters per second

Conversion

694 miles per hour is equivalent to 310.24576 meters per second.[1]

Conversion in the opposite direction

The inverse of the conversion factor is that 1 meter per second is equal to 0.00322325114128876 times 694 miles per hour.

It can also be expressed as: 694 miles per hour is equal to $\frac{1}{\mathrm{0.00322325114128876}}$ meters per second.

Approximation

An approximate numerical result would be: six hundred and ninety-four miles per hour is about three hundred and ten point two five meters per second, or alternatively, a meter per second is about zero times six hundred and ninety-four miles per hour.

Footnotes

[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).

Results may contain small errors due to the use of floating point arithmetic.


text:
## Precalculus (10th Edition)

$x=-26$
Distribute $3$ on th eleft side and $4$ on the right side of the equation to obtain: $3(x-2)=4(x+5)\\ 3x-6=4x+20$ Subtract $20$ from each side: $3x-6-20=4x+20-20\\ 3x-26=4x$ Subtract $3x$ from each side: $3x-26-3x=4x-3x\\ -26=x$


text:
PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Banach Center Publications

1998 | 43 | 1 | 9-24
Tytuł artykułu

### Singleton independence

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Motivated by the central limit problem for algebraic probability spaces arising from the Haagerup states on the free group with countably infinite generators, we introduce a new notion of statistical independence in terms of inequalities rather than of usual algebraic identities. In the case of the Haagerup states the role of the Gaussian law is played by the Ullman distribution. The limit process is realized explicitly on the finite temperature Boltzmannian Fock space. Furthermore, a functional central limit theorem associated with the Haagerup states is proved and the limit white noise is investigated.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
9-24
Opis fizyczny
Daty
wydano
1998
Twórcy
autor
• Centro Vito Volterra, Università di Roma "Tor Vergata", 00133 Roma, Italy
autor
• Graduate School of Polymathematics, Nagoya University, Nagoya 464-8602, Japan
autor
• Graduate School of Polymathematics, Nagoya University, Nagoya 464-8602, Japan
Bibliografia
• [1] L. Accardi, I. Ya. Aref'eva and I. V. Volovich, The master field for half-planar diagrams and free non-commutative random variables, to appear in Quarks `96 (V. Matveev and V. Rubakov, eds.), HEP-TH/9502092.
• [2] L. Accardi, A. Frigerio and J. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto University 18 (1982), 97-133.
• [3] L. Accardi, S. V. Kozyrev and I. V. Volovich, Dynamics of dissipative two-state systems in the stochastic approximation, Phys. Rev. A 56 (1997), 1-7.
• [4] L. Accardi, Y. Hashimoto and N. Obata, Notions of independence related to the free group, Infinite Dimen. Anal. Quantum Probab. 1 (1998), 221-246.
• [5] M. Bożejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987), 170-186.
• [6] M. Bożejko, Positive definite kernels, length functions on groups and noncommutative von Neumann inequality, Studia Math. 95 (1989), 107-118.
• [7] M. Bożejko, Harmonic analysis on discrete groups and noncommutative probability, Volterra preprint series No. 93, 1992.
• [8] M. Bożejko, private communication, November, 1997.
• [9] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Commun. Math. Phys. 185 (1997), 129-154.
• [10] M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), 357-388.
• [11] M. Bożejko and R. Speicher, ψ-Independent and symmetrized white noises, in: Quantum Probability and Related Fields VI, pp. 219-236, World Scientific, 1991.
• [12] I. Chiswell, Abstract length functions in groups, Math. Proc. Camb. Phil. Soc. 80 (1976), 451-463.
• [13] F. Fagnola, A Lévy theorem for free noises, Probab. Th. Rel. Fields 90 (1991), 491-504. %Preprint,1991, Rendiconti Accademia dei Lincei (1992).
• [14] A. Figà-Talamanca and M. Picardello, Harmonic Analysis on Free Groups, Marcel Dekker, New York and Basel, 1983.
• [15] M. de Giosa and Y. G. Lu, From quantum Bernoulli process to creation and annihilation operators on interacting q-Fock space, to appear in Nagoya Math. J.
• [16] N. Giri and W. von Waldenfels, An algebraic version of the central limit theorem, ZW 42 (1978), 129-134.
• [17] U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293.
• [18] Y. Hashimoto, Deformations of the semi-circle law derived from random walks on free groups, to appear in Prob. Math. Stat. 18 (1998).
• [19] F. Hiai and D. Petz, Maximizing free entropy, Preprint No.17, Mathematical Institute, Hungarian Academy of Sciences, Budapest, 1996.
• [20] A. Hora, Central limit theorems and asymptotic spectral analysis on large graphs, submitted to Infinite Dimensional Analysis and Quantum Probability, 1997.
• [21] R. Lenczewski, Quantum central limit theorems, in: Symmetries in Sciences VIII (B. Gruber, ed.), pp. 299-314, Plenum, 1995.
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text:
## The Annals of Applied Probability

### Random interlacements and amenability

#### Abstract

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039–2087] for the special case of ${\mathbb{Z}}^{d}$ (with $d\geq3$). In Sznitman [Ann. of Math. (2) (2010) 171 2039–2087], it was shown that on ${\mathbb{Z}}^{d}$: for any intensity $u>0$, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity $u$ the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of $u$. Finally, we establish the monotonicity of the transition between the “disconnected” and the “connected” phases, providing the uniqueness of the critical value $u_{c}$ where this transition occurs.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 3 (2013), 923-956.

Dates
First available in Project Euclid: 7 March 2013

https://projecteuclid.org/euclid.aoap/1362684850

Digital Object Identifier
doi:10.1214/12-AAP860

Mathematical Reviews number (MathSciNet)
MR3076674

Zentralblatt MATH identifier
1375.60139

#### Citation

Teixeira, Augusto; Tykesson, Johan. Random interlacements and amenability. Ann. Appl. Probab. 23 (2013), no. 3, 923--956. doi:10.1214/12-AAP860. https://projecteuclid.org/euclid.aoap/1362684850

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