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 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 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 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 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

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

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

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

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.

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

We introduce a new concept of statistical solution in the framework of weak solutions to the barotropic Navier–Stokes system with inhomogeneous boundary conditions. Statistical solution is a family \(\{ M_t \}_{t \ge 0}\) of Markov operators on the set of probability measures \(\mathfrak {P}[\mathcal {D}]\) on the data space \(\mathcal {D}\) containing the initial data \([\varrho _0, \mathbf{m}_0]\)

We address the well-posedness of the Cauchy problem corresponding to the relativistic first-order fluid equations, coupled with the Chapman–Enskog heat-flux constitutive relation. We show that the system of equations that results by considering linear perturbations with respect to a generic time direction is non-hyperbolic, since there are modes that may arbitrarily grow as wave-number increases. Then

This article investigates the intersection numbers of the moduli space of p-spin curves with the help of matrix models. The explicit integral representations that are derived for the generating functions of these intersection numbers exhibit p Stokes domains, labelled by a “spin”-component l taking values \(l = -1, 0, 1, 2,...,p-2\). Earlier studies concerned integer values of p, but the present formalism

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in Keller (Attractors and bifurcations of the stochastic

Let z be the activity of point particles described by classical equilibrium statistical mechanics in \(\mathbf{R}^\nu \). The correlation functions \(\rho ^z(x_1,\dots ,x_k)\) denote the probability densities of finding k particles at \(x_1,\dots ,x_k\). Letting \(\phi ^z(x_1,\dots ,x_k)\) be the cluster functions corresponding to the \(\rho ^z(x_1,\dots ,x_k)/z^k\) we prove identities of the type

Starting from the microscopic reduced Hartree–Fock equation, we derive the macroscopic linearized Poisson–Boltzmann equation for the electrostatic potential associated with the electron density.

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with

We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different

Let \(Z^1\) and \(Z^2\) be partition functions in the random polymer model in the same environment but driven by different underlying random walks. We give a comparison in concave stochastic order between \(Z^1\) and \(Z^2\) if one of the random walks has “more randomness” than the other. We also treat some related models: The parabolic Anderson model with space–time Lévy noise; Brownian motion among

We present the complete theory for the decay rates of non-equilibrium fluctuations in a ternary liquid mixture subjected to a stationary temperature gradient, when the quiescent non-convective state is stable. In the most general case, within Boussinesq approximation, four fluctuating modes exist. Depending on the parameter values, propagative modes may be present, and we discuss numerically some cases

We construct marked Gibbs point processes in \({\mathbb {R}}^d\) under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks—attached to the locations in \({\mathbb {R}}^d\)—belong to a general normed space

We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery

In this paper, we firstly construct a weakly coupled Toda lattices with indefinite metrics which consist of 2N different coupled Hamiltonian systems. Afterwards, we consider the iso-spectral manifolds of extended tridiagonal Hessenberg matrix with indefinite metrics which is an extension of a strict tridiagonal matrix with indefinite metrics. For the initial value problem of the extended symmetric

The paper continues (Shcherbina and Shcherbina in Commun Math Phys 351:1009–1044, 2017); Shcherbina in Commun Math Phys 328:45–82, 2014) which study the behaviour of second correlation function of characteristic polynomials of the special case of \(n\times n\) one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix \(J=(-W^2\triangle

We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at 0 of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments \({\gamma }_k=\lim _{n\rightarrow \infty }n^{-[\frac{k}{2}]}{\mathbb {E}}(\sum

Let \(\Phi ^h(x)\) with \(x=(t,y)\) denote the near-critical scaling limit of the planar Ising magnetization field. We take the limit of \(\Phi ^h\) as the spatial coordinate y scales to infinity with t fixed and prove that it is a stationary Gaussian process X(t) whose covariance function K(t) is the Laplace transform of a mass spectral measure \(\rho \) of the relativistic quantum field theory associated

Using an alternative notion of entropy introduced by Datta, the max-entropy, we present a new simplified framework to study the minimizers of the specific free energy for random fields which are weakly dependent in the sense of Lewis, Pfister, and Sullivan. The framework is then applied to derive the variational principle for the loop O(n) model and the Ising model in a random percolation environment

In this paper, we use the Kolmogorov distance to investigate the Smoluchowski–Kramers approximation for stochastic differential equations. We obtain an explicit Berry–Esseen error bound for the rate of convergence. Our main tools are the techniques of Malliavin calculus.

We consider a one-dimensional traffic model with a slow-to-start rule. The initial position of the cars in \(\mathbb {R}\) is a Poisson process of parameter \(\lambda \). Cars have speed 0 or 1 and travel in the same direction. At time zero the speed of all cars is 0; each car waits a mean-one exponential time to switch speed from 0 to 1 and stops when it collides with a stopped car. When the car is

It is shown that the recently proposed quantum analogue of classical energy equipartition theorem for two paradigmatic, exactly solved models (i.e., a free Brownian particle and a dissipative harmonic oscillator) also holds true for all quantum systems which are composed of an arbitrary number of non-interacting or interacting particles, subjected to any confining potentials and coupled to thermostat

We provide a stochastic mathematical representation for Clausius’ and Kelvin-Planck’s statements of the Second Law of Thermodynamics in terms of the entropy productions of a finite, compact driven Markov system and its lift. A surjective map is rigorously established through the lift when the state space is either a discrete graph or a continuous n-dimensional torus \({\mathbb {T}}^n\). The corresponding

In independent bond percolation on \({\mathbb {Z}}^d\) with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for \(d \ge 19\), the set of such p contains values strictly larger than the percolation threshold \(p_c\)

We analyze the hydrodynamic behavior of the porous medium model (PMM) in a discrete space \(\{0,\ldots , n\}\), where the sites 0 and n stand for reservoirs. Our strategy relies on the entropy method of Guo et al. (Commun Math Phys 118:31–59, 1988). However, this method cannot be straightforwardly applied, since there are configurations that do not evolve according to the dynamics (blocked configurations)

We study the nonlinear Schrödinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a

This paper is an attempt to understand time-reversal asymmetry better by developing the quantitative description of that asymmetry. The aim is not to explain the asymmetry, but to describe it in more detail. Two model systems are considered here; one is the classical Lorentz gas, the other a quantum Lorentz gas. In the classical case, it is argued that the distribution of the directions of motion of