We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös–Rényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space

A general framework for the field-theoretic thermodynamic uncertainty relation was recently proposed and illustrated with the \((1+1)\) dimensional Kardar–Parisi–Zhang equation. In the present paper, the analytical results obtained there in the weak coupling limit are tested via a direct numerical simulation of the KPZ equation with good agreement. The accuracy of the numerical results varies with

We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale \(\hbar = N^{-1/3}\) in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition

In the classical parking problem, unit intervals (“car lengths”) are placed uniformly at random without overlapping. The process terminates at saturation, i.e. until no more unit intervals can be stowed. In this paper, we present a generalization of this problem in which the unit intervals are placed with an exponential distribution with rate parameter \(\lambda \). We show that the mathematical expectation

Given a finite family \({\mathcal {U}}\) of finite subsets of \({\mathbb {Z}}^d\setminus \{0\}\), the \({\mathcal {U}}\)-voter dynamics in the space of configurations \(\{+,-\}^{{\mathbb {Z}}^d}\) is defined as follows: every \(v\in {\mathbb {Z}}^d\) has an independent exponential random clock, and when the clock at v rings, the vertex v chooses \(X\in {\mathcal {U}}\) uniformly at random. If the set

We calculate, for a branching random walk \(X_n(l)\) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable \(\frac{1}{2^{n}}\sum _{l=1}^{2^n}e^{2\beta X_n(l)}\), for \(\beta \in {\mathbb {R}}\). We obtain explicit formulae for the first few moments for finite n. In the limit \(n\rightarrow \infty \), our expression coincides with recent conjectures and results

In this paper, we first introduce some new notions of ‘periodic-like’ points, such as almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points, of free semigroup actions. We find that the corresponding sets and gap-sets of these points carry full upper capacity topological entropy of free semigroup actions under certain conditions. Furthermore, \(\phi \)-irregular

We study the dynamics of a Duffing oscillator excited by correlated random perturbations for both fixed and periodically modulated stiffness. In the case of fixed stiffness we see that Poincaré map gets distorted due to the random excitation and, the distortion increases with the increase of correlation of the field. In a strongly correlated field, however, the map becomes purely random. We analyse

We study a concrete model of a confined particle in form of a Schrödinger operator with a compactly supported smooth potential coupled to a bosonic field at positive temperature. We show, that the model exhibits thermal ionization for any positive temperature, provided the coupling is sufficiently small. Mathematically, one has to rule out that zero is an eigenvalue of the self-adjoint generator of

In this paper we define infinite weakly hyperbolic iterated function systems associated with uniformly Dini continuous weight functions. We study the Ruelle operator theorem for the infinite weakly hyperbolic iterated function systems associated with uniformly Dini continuous weight functions. We prove the existence and uniqueness of the invariant measure for these type systems.

In this paper, we confirm the conjecture of Wang and Zhang (J Stat Phys 134 (5-6):953–968, 2009) in a long time scale, i.e., the displacement of the wavefront for 1D nonlinear random Schrödinger equation is of logarithmic order in time |t|.

In this article, we introduce a new approach towards the statistical learning problem \(\mathrm{argmin}_{\rho (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))\) to approximate a target quantum state \(\rho _{\star }\) by a set of parametrized quantum states \(\rho (\theta )\) in a quantum \(L^2\)-Wasserstein metric. We solve this estimation problem by considering Wasserstein

We present rigorous results for the SU(n) Fermi–Hubbard models with finite-range hopping in d (\(\ge 2\)) dimensions. The models are defined on a class of decorated lattices. We first study the models with flat bands at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n) ferromagnetism when the number of particles is equal to the number of unit cells. We then perturb

We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards have a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probabilities (acips) of the slap map

We obtain the exact large deviation functions of the density profile and of the current, in the non-equilibrium steady state of a one dimensional symmetric simple exclusion process coupled to boundary reservoirs with slow rates. Compared to earlier results, where rates at the boundaries are comparable to the bulk ones, we show how macroscopic fluctuations are modified when the boundary rates are slower

We consider the \(d\ (\ge 3)\) - dimensional lattice Gaussian free field on \(\varLambda _N :=[-N, N]^d\cap \mathbb {Z}^d\) in the presence of a self-potential of the form \(U(r)= -b I(|r|\le a)\), \(a>0, b\in \mathbb {R}\). When \(b>0\), the potential attracts the field to the level around zero and is called square-well pinning. It is known that the field turns to be localized and massive for every

We introduce quantum Markov states (QMS) in a general tree graph \(G= (V, E)\), extending the Cayley tree’s case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of the present paper concerns the diagonalizability of a locally faithful QMS \(\varphi \) on a UHF-algebra \({\mathcal {A}}_V\) over the considered tree by means of a suitable conditional

We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered

We introduce the notion of transmission time to study the dynamics of disordered quantum spin chains and prove results relating its behavior to many-body localization properties. We also study two versions of the so-called Local Integrals of Motion (LIOM) representation of spin chain Hamiltonians and their relation to dynamical many-body localization. We prove that uniform-in-time dynamical localization

We study the vertex classification problem on a graph whose vertices are in \(k\ (k\ge 2)\) different communities, edges are only allowed between distinct communities, and the number of vertices in different communities are not necessarily equal. The observation is a weighted adjacency matrix, perturbed by a scalar multiple of the Gaussian Orthogonal Ensemble (GOE), or Gaussian Unitary Ensemble (GUE)

Several theoretical models based on totally asymmetric simple exclusion process (TASEP) have been extensively utilized to study various non-equilibrium transport phenomena. Inspired by the the role of microtubule-transported vesicles in intracellular transport, we propose a generalized TASEP model, where two distinct particles are directed to hop stochastically in opposite directions on a flexible

This study firstly proposes a simple recursive method for deriving the macroscale equations from lattice Boltzmann equations. Similar to the Maxwell iteration based on the convective scaling, this method is used to expand the lattice Boltzmann (LB) equations with the time step \(\delta _{t}\). It is characterised by the incorporation of a nonequilibrium distribution function not appearing in the Maxwell

In this paper we study a multi-species disordered model on the Nishimori line. The typical properties of this line, a set of identities and inequalities among correlation functions, allow us to prove the replica symmetry i.e. the concentration of the order parameter. When the interaction structure is elliptic we rigorously compute the exact solution of the model in terms of a finite-dimensional variational

The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centers of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters, where the rank is assigned to clusters according to the cluster sizes. The distribution function does not depend on boundary conditions and is valid for infinite

We consider multi-dimensional Schrödinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability

We study the existence and uniqueness of the time evolution of a system of infinitely many individuals, moving in a tunnel and subjected to a Cucker–Smale type alignment dynamics with compactly supported communication kernels and to short-range repulsive interactions to avoid collisions.

The Lohe hierarchy is a hierarchy of finite-dimensional aggregation models consisting of the Kuramoto model, the complex Lohe sphere model, the Lohe matrix model and the Lohe tensor model. In contrast, the Schrödinger–Lohe model is the only known infinite-dimensional Lohe aggregation model in literature. In this paper, we provide an explicit connection between the Schrödinger–Lohe model and the complex

The optical conductivity is the basic defining property of materials characterizing the current response toward time-dependent electric fields. In this work, following the approach of Kubo’s response theory, we study the general properties of the nonlinear optical conductivities of quantum many-body systems both in equilibrium and non-equilibrium. We obtain an expression of the second- and the third-order

We correct an incorrect constant in the statement of Theorem 10.6.

Consider the distances \(\tilde{R}_o\) and \(R_o\) from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the d-dimensional Poisson–Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of \(\tilde{R}_o\) and \(R_o\). First, using the well-known relationship between the 0-cell and the typical cell, we show that

The Boltzmann collision operator for long-range interactions is usually employed in its “weak form” in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the principle value sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator

We investigate local statistics of eigenvalues for random normal matrices, represented as 2D determinantal Coulomb gases, in the case when the eigenvalues are forced to be in the support of the equilibrium measure associated with an external field. For radially symmetric external fields with sufficient growth at infinity, we show that the fluctuations of the spectral radius around a hard edge tend

In this article, an inhomogeneous Erdős–Rényi random graph on \(\{1,\ldots , N\}\) is considered, where an edge is placed between vertices i and j with probability \(\varepsilon _N f(i/N,j/N)\), for \(i\le j\), the choice being made independently for each pair. The integral operator \(I_f\) associated with the bounded function f is assumed to be symmetric, non-negative definite, and of finite rank

We discuss what ground states for generic interactions look like. We note that a recent result, due to Morris, implies that the behaviour of ground-state measures for generic interactions is similar to that of generic measures. In particular, it follows from his observation that they have singular spectrum and that they are weak mixing, but not mixing.

Random constraint satisfaction problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of k-uniform hypergraphs with a

We study a 2D system of identical mobile particles on the surface of a cylinder of finite length d and circumference W, immersed in a medium of dielectric constant \(\varepsilon \). The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge \(-e\) (e being the elementary charge) are coined as “counterions”; the system as a whole is

In 1956 Mark Kac proposed a process related to the telegrapher equation where the particle travels at constant speed (say the speed of light c) and randomly inverts its velocity. This process had important applications concerning the path-integral solution and the probabilistic interpretation of the 1\(+\)1 dimensions Dirac equation. The extension to 3\(+\)1 dimensions requires that the particle only

We study linear response theory and entropic fluctuations of finite dimensional non-equilibrium Repeated Interaction Systems (RIS). More precisely, in a situation where the temperatures of the probes can take a finite number of different values, we prove analogs of the Green–Kubo fluctuation–dissipation formula and Onsager reciprocity relations on energy flux observables. Then we prove a Large Deviation

This work concerns the averaging principle for multiscale stochastic fractional Schrödinger–Korteweg-de Vries system. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, the multiscale system can be reduced to a single stochastic fractional Korteweg-de Vries equation

The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix \(G+\mathrm {diag}(\mathbf {a})\), where G is the random matrix from the Gaussian Unitary Ensemble (GUE), and \(\mathrm {diag}(\mathbf {a})\) is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application

We rigorously justify the mean-field limit of an N-particle system subject to Brownian motions and interacting through the Newtonian potential in \({\mathbb {R}}^3\). Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and

We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence b(N) of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the equation \(X_t=x+B_t^H+\int _0^t b(X_s)ds\) where \(\{B_t^H, t\in [0,T]\}\) is a fractional Brownian motion with Hurst parameter \(H\in \big (\frac{1}{2},1\big )\). Based on the techniques of Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence.

We consider the convergence of point-to-point partition functions for the half-space directed polymer model in dimension 1+1 in the intermediate disorder regime as introduced for the full space model by Alberts, Khanin and Quastel in [1]. By scaling the inverse temperature as \(\beta n^{-1/4}\), the point-to-point partition function converges to the chaos series for the solution to stochastic heat

For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowen’s Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that the set of periodic points weighted with the potential equidistributes

A meander system is a union of two arc systems that represent non-crossing pairings of the set \([2n] = \{1, \ldots , 2n\}\) in the upper and lower half-plane. In this paper, we consider random meander systems. We show that for a class of random meander systems,—for simply-generated meander systems,—the number of cycles in a system of size n grows linearly with n and that the length of the largest

We study Random Sequential Adsorption (RSA) of mixture of two-component circular discs on a two-dimensional continuum substrate by computer simulation for different values of radius ratio \({r}_{A}/{r}_{B}\), \(({r}_{A}<{r}_{B})\), and relative rate constant \(k = {k}_{A}/{k}_{B}\) between the discs. For smaller values of radius ratio and all values of relative rate constant between the discs, the

We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given

We consider the storage properties of temporal patterns, i.e. cycles of finite lengths, in neural networks represented by (generally asymmetric) spin glasses defined on random graphs. Inspired by the observation that dynamics on sparse systems has more basins of attractions than the dynamics of densely connected ones, we consider the attractors of a greedy dynamics in sparse topologies, considered

We use asymptotic methods from the theory of differential equations to obtain an analytical expression for the survival probability of an Ornstein–Uhlenbeck process with a potential defined over a broad domain. We form a uniformly continuous analytical solution covering the entire domain by asymptotically matching approximate solutions in an interior region, centered around the origin, to those in

This note discusses the spectral gap of the Fredrickson–Andersen one spin facilitated model in two different settings. The model describes an interacting particle system on a graph, where each site is either occupied or empty; and a site may change its occupation when at least one of its neighbors is empty. We will first consider the model on the infinite lattice \({\mathbb {Z}}^{d}\), with density

In [1], on p. 366, line 5 and on p. 369, line 12, replace

We study the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on \(\mathcal {C}([0,T],\mathscr {P}(\mathbb {R}))\) and all the large

We study binary opinion dynamics in a fully connected network of interacting agents. The agents are assumed to interact according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. We focus on the scenario where

We consider planar stationary exponential last passage percolation in the positive quadrant with boundary weights. For \(\rho \in (0,1)\) and points \(v_N=( (1-\rho )^2 N, \rho ^2 N)\) going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations

In this paper we study the Maki-Thompson rumor model on infinite Cayley trees. The basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals: ignorants, spreaders and stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying

We give general sufficient conditions to prove the convergence of marked point processes that keep record of the occurrence of rare events and of their impact for non-autonomous dynamical systems. We apply the results to sequential dynamical systems associated to both uniformly and non-uniformly expanding maps and to random dynamical systems given by fibred Lasota Yorke maps.

We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we obtain a functional limit theorem to Gaussian vectors. In superdiffusive regime, we obtain strong convergence to a non-Gaussian random vector and characterize its moments

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with slowed reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed