Showing papers in "Journal of Statistical Physics in 2019"

Valency-Based Topological Descriptors and Structural Property of the Generalized Sierpiński Networks

Abstract: A molecular network can be characterized by several different ways, like a matrix, a polynomial, a drawing or a topological descriptor. A topological descriptor is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. Analyzing and determining the topological indices and structural properties of a network or a graph have been a worthy studied topic in the field of chemistry, networks analysis, etc. In this paper, we consider several types of the generalized Sierpinski networks and investigate the explicit expressions of some well-known valency-based topological indices. Taking into account the other structural property of the generalized Sierpinski networks, the average degree is determined.

Delay-coordinate maps and the spectra of Koopman operators

Abstract: The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra.

Synchronization and Stability for Quantum Kuramoto

Abstract: We present and analyze a nonabelian version of the Kuramoto system, which we call the Quantum Kuramoto system. We study the stability of several classes of special solutions to this system, and show that for certain connection topologies the system supports multiple attractors. We also present estimates on the maximal possible heterogeneity in this system that can support an attractor, and study the effect of modifications analogous to phase-lag.

On the Hohenberg-Mermin-Wagner Theorem and Its Limitations

Abstract: Just over 50 years ago, Pierre Hohenberg developed a rigorous proof of the non-existence of long-range order in a two-dimensional superfluid or superconductor at finite temperatures. The proof was immediately extended by Mermin and Wagner to the Heisenberg ferromagnet and antiferromagnet, and shortly thereafter, by Mermin to prove the absence of translational long-range order in a two-dimensional crystal, whether in quantum or classical mechanics. In this paper, we present an extension of the Hohenberg–Mermin–Wagner theorem to give a rigorous proof of the impossibility of long-range ferromagnetic order in an itinerant electron system without spin-orbit coupling or magnetic dipole interactions. We also comment on some situations where there are compelling arguments that long-range order is impossible but no rigorous proof has been given, as well as situations, such as a magnet with long range interactions, or orientational order in a two-dimensional crystal, where long-range order can occur that breaks a continuous symmetry.

The Ising Partition Function: Zeros and Deterministic Approximation

Abstract: We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $$\beta $$ (the interaction) and $$\lambda $$ (the external field), except for the case $$\left| \lambda \right| =1$$ (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all $$\beta ,\lambda $$ , based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

Large deviations conditioned on large deviations I: Markov chain and Langevin equation

Abstract: We present a systematic analysis of stochastic processes conditioned on an empirical observable $$Q_T$$ defined in a time interval [0, T], for large T. We build our analysis starting with a discrete time Markov chain. Results for a continuous time Markov process and Langevin dynamics are derived as limiting cases. In the large T limit, we show how conditioning on a value of $$Q_T$$ modifies the dynamics. For a Langevin dynamics with weak noise and conditioned on $$Q_T$$ , we introduce large deviation functions and calculate them using either a WKB method or a variational formulation. This allows us, in particular, to calculate the typical trajectory and the fluctuations around this trajectory when conditioned on a certain value of $$Q_T$$ , for large T.

Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models

Abstract: We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb–Liniger and Gaudin–Yang models, starting from the Bethe ansatz solution. This makes it possible to calculate explicitly the coefficients of these series and to study their large order behavior. We find that both series diverge factorially and are not Borel summable. In the case of the Gaudin–Yang model, the first Borel singularity is determined by the non-perturbative energy gap. This provides a new perspective on the Cooper instability.

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Abstract: Topological phases protected by symmetry can occur in gapped and—surprisingly—in critical systems. We consider non-interacting fermions in one dimension with spinless time-reversal symmetry. It is known that the phases are classified by a topological invariant $$\omega $$ and a central charge c. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract $$\omega $$ . Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding $$\omega $$ and c. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.

On the Multi-dimensional Elephant Random Walk

Abstract: The purpose of this paper is to investigate the asymptotic behavior of the multi-dimensional elephant random walk (MERW). It is a non-Markovian random walk which has a complete memory of its entire history. A wide range of literature is available on the one-dimensional ERW. Surprisingly, no references are available on the MERW. The goal of this paper is to fill the gap by extending the results on the one-dimensional ERW to the MERW. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the MERW. The asymptotic normality of the MERW, properly normalized, is also provided. In the superdiffusive regime, we prove the almost sure convergence as well as the mean square convergence of the MERW. All our analysis relies on asymptotic results for multi-dimensional martingales.

A Kinetic Model for a Polyatomic Gas with Temperature-Dependent Specific Heats and Its Application to Shock-Wave Structure

Abstract: The ellipsoidal statistical (ES) model of the Boltzmann equation for a polyatomic gas with constant specific heats (calorically perfect gas), proposed by Andries et al. (Eur J Mech B Fluids 19:813, 2000), is extended to a polyatomic gas with temperature-dependent specific heats (thermally perfect gas). Then, the new model equation is applied to investigate the structure of a plane shock wave with special interest in $$\hbox {CO}_2$$ gas, which is known to have a very large bulk viscosity, and in the case of relatively strong shock waves. A numerical analysis, as well as an asymptotic analysis for large bulk viscosity, is performed in parallel to the previous paper by two of the present authors (Kosuge and Aoki, in: Phys Rev Fluids 3:023401, 2018), where the structure of a shock wave in $$\hbox {CO}_2$$ gas was investigated using the ES model for a polyatomic gas with constant specific heats. From the numerical and analytical results, the effect of temperature-dependent specific heats on the structure of a shock wave is clarified.

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Abstract: Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability $$\alpha $$ the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits a phase transition from diffusive to superdiffusive behavior at the critical value $$\alpha _c=1/2$$. For $$\alpha \in (\alpha _c, 1)$$, there is a scaling factor $$a_n$$ of order $$n^{\alpha }$$ such that the position $$S_n$$ of the walker at time n scaled by $$a_n$$ converges to a nondegenerate random variable $${\widehat{W}}$$, whose distribution is not Gaussian. Our main result shows that the fluctuation of $$S_n$$ around $${\widehat{W}} \cdot a_n$$ is still Gaussian. We also give a description of a phase transition induced by bias decaying polynomially in time.

Effective Fluctuation and Response Theory

Abstract: The so-called fluctuation theorems pushed the study of systems far beyond equilibrium, whose response to thermodynamic forces (affinities) is characterized by the reciprocal and the fluctuation-dissipation relations. All these results rely on the assumption that the observer has complete information about the system: no hidden leakage to the environment, exact evaluation of the thermodynamic cost of processes. Will an observer who has marginal information be able to perform an effective thermodynamic analysis? Given that such observer will only be able to establish local equilibrium, by perturbing the stalling currents will he/she observe equilibrium-like fluctuations? Within the formalism of Markov jump processes on finite networks, we provide a broad theory of the statistical behavior of some out of many currents that flow across a system. In particular (1) There exist effective affinities for which an integral fluctuation relation holds; (2) At stalling, i.e. where the marginal currents vanish, a symmetrized fluctuation-dissipation relation holds; (3) Under reasonable assumptions on the parametrization of the rates, effective affinities can be operationally defined by a procedure of tuning to stalling; (4) There exists a notion of marginally time-reversed process which restores the full-fledged fluctuation relation and reciprocity; (5) There exist fluctuation relations across different levels in the hierarchy of more and more “complete” theories. The above results apply to configuration-space currents, and to their phenomenological linear combinations provided certain symmetries of the effective affinities are respected—a condition whose range of validity we deem the most interesting question left open to future inquiry.

Interacting Diffusions on Random Graphs with Diverging Average Degrees: Hydrodynamics and Large Deviations

Abstract: We consider systems of mean-field interacting diffusions, where the pairwise interaction structure is described by a sparse (and potentially inhomogeneous) random graph. Examples include the stochastic Kuramoto model with pairwise interactions given by an Erdős–Renyi graph. Our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions. For a broad class of interaction functions, we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit, which is given by a McKean–Vlasov diffusion. Moreover, we also prove matching behavior of the two systems at the level of large deviations. Our results extend classical results of dai Pra and den Hollander and provide the first examples of LDPs for systems with sparse random interactions.

Particle and Kinetic Models for Swarming Particles on a Sphere and Stability Properties

Abstract: We present particle and kinetic models for the description of swarming particles on a sphere in the presence of random noises, and study their stability properties. In the absence of noises, the proposed particle model can be reduced from the Lohe matrix model for quantum synchronization, and the kinetic model can be formally derived from the particle swarming model using the BBGKY hierarchy. For the particle model without noises, we show that it is uniformly stable with respect to initial data in a Lebesgue norm. This uniform stability and particle-in-cell method yield a global existence of a measure-valued solution to the corresponding inviscid kinetic model. We also show that the incoherent state for the kinetic model is nonlinearly stable, as long as the ratio between noise strength and coupling strength is sufficiently large.

Experimental Study of the Bottleneck in Fully Developed Turbulence

Abstract: The energy spectrum of incompressible turbulence is known to reveal a pileup of energy at those high wavenumbers where viscous dissipation begins to act. It is called the bottleneck effect (Donzis and Sreenivasan in J Fluid Mech 657:171–188, 2010; Falkovich in Phys Fluids 6:1411–1414, 1994; Frisch et al. in Phys Rev Lett 101:144501, 2008; Kurien et al. in Phys Rev E 69:066313, 2004; Verma and Donzis in Phys A: Math Theor 40:4401–4412, 2007). Based on direct numerical simulations of the incompressible Navier-Stokes equations, results from Donzis and Sreenivasan (657:171–188, 2010) pointed to a power-law decrease of the strength of the bottleneck with increasing intensity of the turbulence, measured by the Taylor micro-scale Reynolds number $$R_{\lambda }$$ . Here we report the first experimental results on the dependence of the amplitude of the bottleneck as a function of $$R_{\lambda }$$ in a wind-tunnel flow. We used an active grid (Griffin et al. in Control of long-range correlations in turbulence, arXiv:1809.05126 , 2019) in the variable density turbulence tunnel (VDTT) (Bodenschatz et al. in Rev Sci Instrum 85:093908, 2014) to reach $$R_{\lambda }>5000$$ , which is unmatched in laboratory flows of decaying turbulence. The VDTT with the active grid permitted us to measure energy spectra from flows of different $$R_{\lambda }$$ , with the small-scale features appearing always at the same frequencies. We relate those spectra recorded to a common reference spectrum, largely eliminating systematic errors which plague hotwire measurements at high frequencies. The data are consistent with a power law for the decrease of the bottleneck strength for the finite range of $$R_{\lambda }$$ in the experiment.

Large Deviations Conditioned on Large Deviations II: Fluctuating Hydrodynamics

Abstract: For diffusive many-particle systems such as the SSEP (symmetric simple exclusion process) or independent particles coupled with reservoirs at the boundaries, we analyze the density fluctuations conditioned on the current integrated over a large time. We determine the conditioned large deviation function of the density by a microscopic calculation. We then show that it can be expressed in terms of the solutions of Hamilton–Jacobi equations, which can be written for general diffusive systems using a fluctuating hydrodynamics description.

Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation

Abstract: In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant.

Long-Range Order, “Tower” of States, and Symmetry Breaking in Lattice Quantum Systems

Abstract: In a quantum many-body system where the Hamiltonian and the order operator do not commute, it often happens that the unique ground state of a finite system exhibits long-range order (LRO) but does not show spontaneous symmetry breaking (SSB). Typical examples include antiferromagnetic quantum spin systems with Neel order, and lattice boson systems which exhibits Bose–Einstein condensation. By extending and improving previous results by Horsch and von der Linden and by Koma and Tasaki, we here develop a fully rigorous and almost complete theory about the relation between LRO and SSB in the ground state of a finite system with continuous symmetry. We show that a ground state with LRO but without SSB is inevitably accompanied by a series of energy eigenstates, known as the “tower” of states, which have extremely low excitation energies. More importantly, we also prove that one gets a physically realistic “ground state” by taking a superposition of these low energy excited states.

Random Band Matrices in the Delocalized Phase, II: Generalized Resolvent Estimates

Abstract: This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $$N\times N$$ random band matrices $$H=(H_{ij})$$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $$\mathbb {E} |H_{ij}|^2$$ form a band matrix with typical band width $$1\ll W\ll N$$ . We consider the generalized resolvent of H defined as $$G(Z):=(H - Z)^{-1}$$ , where Z is a deterministic diagonal matrix such that $$Z_{ij}=\left( z\mathbb {1}_{1\leqslant i \leqslant W}+\widetilde{z}\mathbb {1}_{ i > W} \right) \delta _{ij}$$ , with two distinct spectral parameters $$z\in \mathbb {C}_+:=\{z\in \mathbb {C}:{{\,\mathrm{Im}\,}}z>0\}$$ and $$\widetilde{z}\in \mathbb {C}_+\cup \mathbb {R}$$ . In this paper, we prove a sharp bound for the local law of the generalized resolvent G for $$W\gg N^{3/4}$$ . This bound is a key input for the proof of delocalization and bulk universality of random band matrices in Bourgade et al. ( arXiv:1807.01559 , 2018). Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in Yang and Yin ( arXiv:1807.02447 , 2018).

Emergent Behavior of a Second-Order Lohe Matrix Model on the Unitary Group

Abstract: We study a second-order extension to the first-order Lohe matrix model on the unitary group which can be reduced to the second-order Kuramoto model with inertia as a special case. For the proposed second-order model, we present several sufficient frameworks leading to the emergence of the complete and practical synchronizations in terms of the initial data and the system parameters. For the identical hamiltonians, we show that the complete synchronization emerges asymptotically. In contrast, for the non-identical hamiltonians, the practical synchronization occurs for some class of initial data when the product of the coupling strength and inertia is sufficiently small.

Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes

Abstract: Among the predictive hidden Markov models that describe a given stochastic process, the $$\epsilon \text{-machine }$$ is strongly minimal in that it minimizes every Renyi-based memory measure. Quantum models can be smaller still. In contrast with the $$\epsilon \text{-machine }$$ ’s unique role in the classical setting, however, among the class of processes described by pure-state hidden quantum Markov models, there are those for which there does not exist any strongly minimal model. Quantum memory optimization then depends on which memory measure best matches a given problem’s circumstance.

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Abstract: We establish the strong law of large numbers for Betti numbers of random Cech complexes built on $${\mathbb {R}}^N$$ -valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on $${\mathbb {R}}^N$$ and the case where it is supported on a $$C^1$$ compact manifold of dimension strictly less than N. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to $$L^p$$ spaces, for all $$1\le p < \infty $$ .

Instability, Rupture and Fluctuations in Thin Liquid Films: Theory and Computations.

Abstract: Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grun et al. (J Stat Phys 122(6):1261–1291, 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. This scheme seems to be very convenient for numerical studies of the stochastic thin-film equation, since it makes it easier to select the frequency modes of the noise (following the spirit of the long-wave approximation). With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies.

Kinetic Modeling of Alcohol Consumption

Abstract: In most countries, alcohol consumption distributions have been shown to possess universal features. Their unimodal right-skewed shape is usually modeled in terms of the Lognormal distribution, which is easy to fit, test, and modify. However, empirical distributions often deviate considerably from the Lognormal model, and both Gamma and Weibull distributions appear to better describe the survey data. In this paper we explain the appearance of these distributions by means of classical methods of kinetic theory of multi-agent systems. The microscopic variation of alcohol consumption of agents around a universal social accepted value of consumption, is built up introducing as main criterion for consumption a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. The mathematical properties of the value function then determine the unique macroscopic equilibrium which results to be a generalized Gamma distribution. The modeling of the microscopic kinetic interaction allows to clarify the meaning of the various parameters characterizing the generalized Gamma equilibrium.

Orthogonal Stochastic Duality Functions from Lie Algebra Representations.

Abstract: We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between $$*$$ -representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and $$\mathfrak {su}(1,1)$$ . Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.

Recursion for the Smallest Eigenvalue Density of β -Wishart-Laguerre Ensemble

Abstract: The statistics of the smallest eigenvalue of Wishart–Laguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost significance in understanding the spectral behavior of Wishart–Laguerre ensembles and, among other things, unveil the underlying universality aspects in the asymptotic limits. However, obtaining exact and explicit expressions by expanding determinants or Pfaffians becomes impractical if large dimension matrices are involved. For the real matrices ( $$\beta =1$$ ) Edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices. Very recently, an analogous recurrence scheme has been obtained for the complex matrices ( $$\beta =2$$ ). In the present work we extend this to $$\beta $$ -Wishart–Laguerre ensembles for the case when exponent $$\alpha $$ in the associated Laguerre weight function, $$\lambda ^\alpha e^{-\beta \lambda /2}$$ , is a non-negative integer, while $$\beta $$ is positive real. This also gives access to the smallest eigenvalue density of fixed trace $$\beta $$ -Wishart–Laguerre ensemble, as well as moments for both cases. Moreover, comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly. Exact evaluations for large values of n (the matrix dimension) and $$\alpha $$ also enable us to compare with Tracy–Widom density and large deviation results of Katzav and Castillo. We also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the Wigner–Smith matrix and are relevant to the problem of quantum chaotic scattering.

Harmonic Analysis in Phase Space and Finite Weyl-Heisenberg Ensembles.

Abstract: Weyl–Heisenberg ensembles are translation-invariant determinantal point processes on $$\mathbb {R}^{2d}$$ associated with the Schrodinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl–Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with N points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area N, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain $$\Omega $$ . We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of $$\Omega $$ , as $$\Omega $$ is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.

The High Temperature Crossover for General 2D Coulomb Gases

Abstract: We consider N particles in the plane, influenced by a general external potential, that are subject to the Coulomb interaction in two dimensions at inverse temperature $$\beta $$ . At large temperature, when scaling $$\beta =2c/N$$ with some fixed constant $$c>0$$ , in the large-N limit we observe a crossover from Ginibre’s circular law or its generalisation to the density of non-interacting particles at $$\beta =0$$ . Using Ward identities and saddle point methods we derive a partial differential equation of generalised Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential as well, and present the analytic solution for the density in the case of a Gaussian plus logarithmic potential.

Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

Abstract: We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of free-fermion vertex weights. We describe how the recently developed ‘Tangent method’ can be used to determine the form of the arctic curve. The obtained result is in agreement with numerics.

Flocking With Short-Range Interactions

Abstract: We study the large-time behavior of continuum alignment dynamics based on Cucker–Smale (CS)-type interactions which involve short-range kernels, that is, communication kernels with support much smaller than the diameter of the crowd. We show that if the amplitude of the interactions is larger than a finite threshold, then unconditional hydrodynamic flocking follows. Since we do not impose any regularity nor do we require the kernels to be bounded, the result covers both regular and singular interaction kernels.Moreover, we treat initial densities in the general class of compactly supported measures which are required to have positive mass on average (over balls at small enough scale), but otherwise vacuum is allowed at smaller scales. Consequently, our arguments of hydrodynamic flocking apply, mutatis mutandis, to the agent-based CS model with finitely many Dirac masses. In particular, discrete flocking threshold is shown to depend on the number of dense clusters of communication but otherwise does not grow with the number of agents.