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

There is an error in the proof of Lemma 4.1.

One dimensional pinning models have been widely studied in the physical and mathematical literature, also in presence of disorder. Roughly speaking, they undergo a transition between a delocalized phase and a localized one. In mathematical terms these models are obtained by modifying the distribution of a discrete renewal process via a Boltzmann factor with an energy that contains only one body potentials

We consider the bipartite Potts model in which interaction strengths depend only on the groups the particles belong to. We first rigorously compute the exact value of the free energy, then put forward two scaling limit theorems for the joint distribution of two empirical vectors which serve as order parameters. It turns out that the limit distribution can be Gaussian or the exponential distribution

Given a classical channel—a stochastic map from inputs to outputs—the input can often be transformed into an intermediate variable that is informationally smaller than the input. The new channel accurately simulates the original but at a smaller transmission rate. Here, we examine this procedure when the intermediate variable is a quantum state. We determine when and how well quantum simulations of

Erratum: Finite-Velocity Diffusion in Random Media. [J Stat Phys 179, 729–747 (2020)]

The maximum likelihood state, which corresponds to the global maxima of the probability density function, is of interest in a variety of applications. The maximum likelihood trajectory is introduced in this paper to describe how the maximum likelihood state of a nonequilibrium system evolves with time during sate transition. Analytical expressions, as well as numerical methods, for the maximum likelihood

Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of \((\kappa , \lambda )\)-capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker

We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions.

For n + 1 particles moving independently on a straight line, we study the question of how long the leading position of one of them can last. Our focus is the asymptotics of the probability pT,n that the leader time will exceed T when n and T are large. It is assumed that the dynamics of particles are described by independent, either stationary or self-similar, Gaussian processes, not necessarily identically

I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products \(\varPi _n=M_nM_{n-1}\ldots M_1\), where \(M_i\)’s are i.i.d. Following Tutubalin (Theor Probab Appl 10(1):15–27, 1965), the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of

We study the statistics of last-passage time for linear diffusions. First we present an elementary derivation of the Laplace transform of the probability density of the last-passage time, thus recovering known results from the mathematical literature. We then illustrate them on several explicit examples. In a second step we study the spectral properties of the Schrödinger operator associated to such

We study the dynamics of a particle moving in a square two-dimensional Lorentz lattice-gas. The underlying lattice-gas is occupied by two kinds of rotators, “right-rotator (R)” and “left-rotator (L)” and some of the sites are empty viz. vacancy “V”.The density of R and L are the same and density of V is one of the key parameters of our model. The rotators deterministically rotate the direction of a

Generating an initial condition for a Langevin equation with memory is a non trivial issue. We introduce a generalisation of the Laplace transform as a useful tool for solving this problem, in which a limit procedure may send the extension of memory effects to arbitrary times in the past. This method allows us to compute average position, work, their variances and the entropy production rate of a particle

We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at \(x=0\). The boundary condition \(\partial _x h(x,t)|_{x=0}=A\) corresponds to an attractive wall for \(A<0\), and leads to the binding of the polymer to the

We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where they are immediately absorbed. It is well-known that the population survives forever with positive probability if and only if the branching rate is sufficiently large. We show that throughout this super-critical

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display

We study the metastable minima of the Curie–Weiss Potts model with three states, as a function of the inverse temperature, and for arbitrary vector-valued external fields. Extending the classic work of Ellis and Wang (Stoch Process Appl 35(1):59–79, 1990) and Wang (Stoch Process Appl 50(2):245–252, 1994) we use singularity theory to provide the global structure of metastable (or local) minima. In particular

We present an extension of the cell-construction method for the flat-band ferromagnetism. In a rather general setting, we construct Hubbard models with highly degenerate single-electron ground states and obtain a formal representation of these single-electron ground states. By our version of the cell-construction method, various types of flat-band Hubbard models, including the one on line graphs, can

We consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states, for instance, between certain microcanonical, canonical, and grand canonical ensemble expectations. A particular goal is to obtain good bounds for how such errors will decay with increasing system size. As explicit examples

We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary,

We analyze the Casimir forces for an ideal Bose gas enclosed between two infinite parallel walls separated by the distance D. The walls are characterized by the Dirichlet boundary conditions. We show that if the thermodynamic state with Bose–Einstein condensate present is correctly approached along the path pertinent to the Dirichlet b.c. then the leading term describing the large-distance decay of

The temporal evolution of Kuramoto oscillators influenced by the temperature field often appears in biological oscillator ensembles. In this paper, we propose a generalized Kuramoto type lattice model on a regular ring lattice with the equal spacing assuming that each oscillator has an internal energy (temperature). Our lattice model is derived from the thermodynamical Cucker–Smale model for flocking

We consider the two-dimensional random matching problem in \({\mathbb {R}}^2.\) In a challenging paper, Caracciolo et al. Phys Rev E 90(1):012118 (2014), on the basis of a subtle linearization of the Monge-Ampère equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is \(\log

We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional

This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Schütz and Trimper (Phys Rev E 70:044510(R), 2004) with stronger reinforcement mechanisms, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability p. With probability

We extend results on quadratic pressure and convergence of Gibbs measures from Leplaideur and Watbled (Bull Soc Math France 147(2):197–219, 2019) to some general models for spin spaces. We define the notion of equilibrium state for the quadratic pressure and show that under some conditions on the maxima for some auxiliary function, the Gibbs measure converges to a convex combination of eigen-measures

Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm

A generalisation of Kingman’s model of selection and mutation has been made in a previous paper which assumes all mutation probabilities to be i.i.d.. The weak convergence of fitness distributions to a globally stable equilibrium was proved. The condensation occurs if almost surely a positive proportion of the population travels to and condensates on the largest fitness value due to the dominance of

We introduce generalized dynamical pruning on rooted binary trees with edge lengths that encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The pruning removes parts of a tree T, starting from the leaves, according to a pruning function defined on descendant subtrees within T. We prove the invariance of critical binary Galton–Watson tree

We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure p-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting

We present a complete framework for determining the asymptotic (or logarithmic) efficiency of estimators of large deviation probabilities and rate functions based on importance sampling. The framework relies on the idea that importance sampling in that context is fully characterized by the joint large deviations of two random variables: the observable defining the large deviation probability of interest

We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere \({\mathbb {S}}^2\) by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on \({\mathbb {S}}^2\). In particular we derive conformal Ward identities for these correlation functions. This forms the basis for

We prove the Gibbs variational formula in terms of quantum relative entropy density that characterizes translation invariant thermal equilibrium states in quantum lattice systems. It is a natural quantum extension of the similar statement established by Föllmer for classical systems.

This paper deals with the asymptotic behavior of solution to the linearized ellipsoidal BGK model in torus. We prove that the solution converges exponentially to the equilibrium in the weighted Sobolev spaces with polynomial weight. Our exponential decay rate \(e^{-\lambda t}\) is optimal in the sense that \(\lambda >0\) equals to the spectral gap of the linearized operator in the standard Hilbert

We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the \(\ell ^\infty \)-category initial state so that the internal graph receives time independent input from the tails, say \(\varvec{\alpha }_{in}\), at every time step. We show that the response of the Szegedy walk to

We consider the kinetic theory of dilute gases in the Boltzmann–Grad limit. We propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann molecular chaos hypothesis (Stosszahlansatz), we derive a large deviation rate function, or action, that describes the stochastic process for the empirical distribution. The quasipotential

During the formation process of stochastic networks, nodes tend to establish edges based on selective linkage mechanisms. In general, these mechanisms involve probability distributions that underlie the selection of target nodes. In social networks, edges are often associated to relationships that are homophilic with respect to individual traits. Such traits include, for example, gender and age, and

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean–Vlasov equation depending on \(\varepsilon \), the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle with a rate functional

We study an interacting particle system in which moving particles activate dormant particles linked by the components of critical bond percolation. Addressing a conjecture from Beckman, Dinan, Durrett, Huo, and Junge for a continuous variant, we prove that the process can reach infinity in finite time i.e., explode. In particular, we prove that explosions occur almost surely on regular trees as well

The spatially homogeneous BGK equation is obtained as the limit of a model of a many particle system, similar to Mark Kac’s charicature of the spatially homogeneous Boltzmann equation.

We consider a minimal model of one-dimensional discrete-time random walk with step-reinforcement, introduced by Harbola, Kumar, and Lindenberg (2014): The walker can move forward (never backward), or remain at rest. For each \(n=1,2,\ldots \), a random time \(U_n\) between 1 and n is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time \(U_n\), then at time \(n+1\) it

We introduce the special issue on the Statistical Mechanics of Climate by presenting an informal discussion of some theoretical aspects of climate dynamics that make it a topic of great interest for mathematicians and theoretical physicists. In particular, we briefly discuss its nonequilibrium and multiscale properties, the relationship between natural climate variability and climate change, the different

In the setting of the fractional quantum Hall effect we study the effects of strong, repulsive two-body interaction potentials of short range. We prove that Haldane’s pseudo-potential operators, including their pre-factors, emerge as mathematically rigorous limits of such interactions when the range of the potential tends to zero while its strength tends to infinity. In a common approach the interaction

Periodic phenomena such as oscillation have been studied for many years. In this paper, we verify the stochastic version of Levinson’s conjecture, which confirmed the existence of stochastic periodic solutions for second order Newtonian systems with dissipativeness. First, we provide a stochastic Duffing’s equation to display our result. Then, we apply Wong–Zakai approximation method and Lyapunov’s

We analyze ensembles of random networks with fixed numbers of edges, triangles, and nodes. In the limit as the number of nodes goes to infinity, this model is known to exhibit sharp phase transitions as the density of edges and triangles is varied. In this paper we study finite size effects in ensembles where the number of nodes is between 30 and 66. Despite the small number of nodes, the phases and

The attractor Hausdorff dimension is an important quantity bridging information theory and dynamical systems, as it is related to the number of effective degrees of freedom of the underlying dynamical system. By using the link between extreme value theory and Poincaré recurrences, it is possible to estimate this quantity from time series of high-dimensional systems without embedding the data. In general

An algorithm is developed for determining the potential barrier height experimentally, provided that we have control over the noise strength \(\sigma \). We are concerned with the situation when the laboratory or numerical experiment requires large resources of time or computational power, respectively, and wish to find a protocol that provides the best estimate in a given amount of time. The optimal

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass \(\mu \) in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity \(\epsilon >0\). We show that for all small but fixed frictions the small mass limit of \(q_{\mu , \epsilon }\) gives the solution \(q_\epsilon

The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to G, with each edge being replaced by a unit resistor. In this study, we apply the principle of substitution to three-towers Hanoi graph to procure an equivalent network that enables us to only use the series and parallel

Recent work has found that the behavior of an individual can be altered when infected by a parasite. Here we explore the question: under what conditions, in principle, can a general parasitic infection control system-wide social behaviors? We analyze fixed points and hysteresis effects under the Master Equation, with transitions between two behaviors given two different subpopulations, healthy vs.

Most studies on the problem of equilibration of the Fermi–Pasta–Ulam–Tsingou (FPUT) system have focused on equipartition of energy being attained amongst the normal modes of the corresponding harmonic system. In the present work, we instead discuss the equilibration problem in terms of local variables, and consider initial conditions corresponding to spatially localized energy. We estimate the time-scales