Beijing Institute of Mathematical Sciences and Applications

The derivation of macroscopic behavior of energy, temperature, density, from microscopic dynamics under a suitable scaling of space and time is at the center of the mathematical research in non-equilibrium statistical mechanics. I will review some recent results on simple microscopic dynamics about diffusive behaviour of heat (Fourier Law) and on the conversion of work into heat. I will expose also an analysis of different boundary conditions, dynamics with more conserved quantities (energy, momentum, density of particles),anomalous transport due to superdiffusion, and thermal isolation in disordered systems. These are joint works with Tomasz Komorowski, Joel Lebowitz, Marielle Simon, Pedro Garrido.

We review recent methods, based on entropy estimates, to derive stochastic partial differential equations, mostly linear, as the limit of the fluctuation fields of interacting particle systems. We apply the method to a stirring dynamics perturbed by a voter model whose fluctuation field converges to the stochastic heat equation.

We review recent methods, based on entropy estimates, to derive stochastic partial differential equations, mostly linear, as the limit of the fluctuation fields of interacting particle systems. We apply the method to a stirring dynamics perturbed by a voter model whose fluctuation field converges to the stochastic heat equation.

The derivation of macroscopic behavior of energy, temperature, density, from microscopic dynamics under a suitable scaling of space and time is at the center of the mathematical research in non-equilibrium statistical mechanics. I will review some recent results on simple microscopic dynamics about diffusive behaviour of heat (Fourier Law) and on the conversion of work into heat. I will expose also an analysis of different boundary conditions, dynamics with more conserved quantities (energy, momentum, density of particles),anomalous transport due to superdiffusion, and thermal isolation in disordered systems. These are joint works with Tomasz Komorowski, Joel Lebowitz, Marielle Simon, Pedro Garrido.

The derivation of macroscopic behavior of energy, temperature, density, from micro-scopic dynamics under a suitable scaling of space and time is at the center of the mathe-matical research in non-equilibrium statistical mechanics. I will review some recent results on simple microscopic dynamics about diffusive behaviour of heat (Fourier Law) and on the conversion of work into heat. I will expose also an analysis of different boundary condi-tions, dynamics with more conserved quantities (energy, momentum, density of particles),anomalous transport due to superdiffusion, and thermal isolation in disordered systems. These are joint works with Tomasz Komorowski, Joel Lebowitz, Marielle Simon, Pedro Garrido.

We review recent methods, based on entropy estimates, to derive stochastic partial differential equations, mostly linear, as the limit of the fluctuation fields of interacting particle systems. We apply the method to a stirring dynamics perturbed by a voter model whose fluctuation field converges to the stochastic heat equation.

We review recent methods, based on entropy estimates, to derive stochastic partial differential equations, mostly linear, as the limit of the fluctuation fields of interacting particle systems. We apply the method to a stirring dynamics perturbed by a voter model whose fluctuation field converges to the stochastic heat equation.

I talk about the stochastic dynamics of infinite particle systems with very strong cor-relations. The equilibrium states of the dynamics classify the systems. The equilibrium states are random point fields ($RPFs$) in $\mathbb{R}^{d}$. Three categories of $RPFs$ exist (1) potential type $RPFs$, (2) kernel type $RPFs$, and (3) zero points of random analytic functions. The typical examples of these systems are Coulomb and $Riesz$ potentials for (1), determinantal $RPFs$ for (2), and the zero points of the planar Gaussian analytic function (PGAF) for(3). I construct stochastic dynamics of these examples in (1), (2), and (3), and present representations through infinite dimensional stochastic differential equations for (1) and the PGAF in (3). For the two dimensional Coulomb systems and the PGAF, I prove the sub-diffusivity of the tagged particles. This result is a phenomenon arising from the strong correlation of the systems.

In this talk, we consider the asymmetric exclusion process on the $1-d$ lattice of size $N$.The system exchanges particles with reservoirs at all sites, with rates that grow rapidly near the boundaries. At the hyperbolic space-time scale, the macroscopic mass density evolves with the entropy solution to a boundary-driven quasilinear conservation law with a relaxation term. Different from the usual first order quasilinear equations with boundary conditions [Bardos, LeRoux, N ́ed ́elec, CPDE, 1979], the solution is continuous in a weak sense at the boundaries due to the strong relaxation scheme. The talk is based on joint works with Linjie Zhao, Patric ́ıa Gon ̧calves and Julian Kern(arXiv 2310.11415, arXiv 2402.11921 and work in progress).

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar, and Lindenberg (2014). Depending on the memory parameter p and the power law exponent β , we obtain three distinct phases. In one such phase the elephant travels only a finite distance almost surely, and the other two phases are distinguished by the speed at which the elephant travels. This talk is based on the joint work with Rahul Roy (Indian Statistical Institute) and Masato Takei (Yokohama National University).

In this talk, we give a method which guarantees a density depended lower bound estimate of the spectral gap for zero-range process by means of Yau’s martingale method. In particular, we give some sufficient conditions and examples.

In this talk, we introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space $\{0,1\}^{\mathbb{Z}^d}$. In our model, a voter adopts one of its neighbors' opinion at rate one except for neighbors crossing the hyperplane $\{x:x_1 = 1/2\}$, where the rate is $\alpha N^{-\beta}$ and thus is called a slow membrane. Above, $\alpha>0,\,\beta \geq 0$ are given parameters. We consider the limit $N \rightarrow \infty$, and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of $\beta$. We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein-Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation. This talk is based on joint work with Linjie Zhao.

The Karkar--Parisi--Zhang universality is believed to describe the limiting behavior of many random surface growth models arising from interacting particle systems, first-/last-passage percolation, directed polymers, etc. The associated 1:2:3 KPZ scaling, confirmed rigorously by the study of exactly solvable models, is a unique feature of the KPZ models which is already quite different from the usual Gaussian 1/2 exponent. The KPZ scaling can also be understood via the large-scale behavior of geometric objects like minimizers and polymer measures in random environments. The stochastic Burgers equation (and Hamilton--Jacobi equations as its generalization) is a KPZ model which connects these geometric objects with the ergodic theory of SPDEs. The variational representation of the solutions can be interpreted as LPP or directed polymer models; the large-time limits of the minimizers and polymer measures are related to the invariant measures of the SPDE. Common features shared by the Burgers, the minimizers and the polymer measures such as monotonicity and contraction, should help us better understand the KPZ scaling and universality.

We focus on the finite-volume approximation rate of diffuision matrix/conductivity in non-gradient Kawasaki Dynamics. The proof mainly follows the quantitative homogenization theory developed by Armstrong, Kuusi, Mourrat and Smart, while the hard-core exclusion rule brings difficulties. We use a new coarse-grained coupling method to deduce important functional inequalities, which are applied to give a proof of the polynomial convergence rate. As an application, our result also imply a convergence rate in the hydrodynamic limit. Based on joint work with Tadahisa Funaki (BIMSA) and Chenlin Gu (Tsinghua).

We consider singular quasilinear stochastic $PDEs$ with spatial white noise as its potential on 1-dimensional torus. Such singular stochastic $PDEs$ are relative to a study of the hydrodynamic scaling limit of a microscopic interacting particle system in a random environment. Under sufficient conditions on coefficients and noise, we study the global existence of solutions in paracontrolled sense, and we also show the convergence of the solutions to the stationary solutions as time goes to infinity. To prove the main results, the proper energy functional is introduced and the approach based on energy inequality and Poincar ́e inequality is used. This talk is based on the joint work with T. Funaki.

We derive the quantitative estimates of propagation of chaos for the large interacting particle systems in terms of the relative entropy between the joint law of the particles and the tensorized law of the mean field PDE. We resolve this problem for the first time for the viscous vortex model that approximating 2D Navier-Stokes equation in the vorticity formulation on the whole space. We obtain as key tools the Li-Yau-type estimates and Hamilton-type heat kernel estimates for 2D Navier-Stokes on the whole space. This is based on a joint work with Xuanrui Feng from Peking University.

We consider the sharp interface limit problem for 1D stochastic Allen-Cahn equation driven by space-time white noise, and extend the classic work by Funaki to the full small noise regime. One interesting point is that it turns out the notion of “small noise” depends on the topology one puts on the deterministic part of the equation. In addition to the functional framework by Funaki, the main new idea is the construction of a series of functional correctors, which are designed to cancel potential divergences recursively. Joint work with Wenhao Zhao (EPFL) and Shuhan Zhou (PKU).

We consider a simple random walk on a b-ary tree of finite depth and study its local time field on the leaves. Biskup and Louidor recently established the randomly shifted Gumbel fluctuation for the maximal local time. In this talk, we will describe a result on the convergence of the extremal process associated with the local time field to a decorated Cox process. This talk is based on ongoing joint work with Marek Biskup.

We prove that the scaling limit of the continuous solid-on-solid model in ℤd is a multiple of the Gaussian free field.

We consider the reversible inclusion process on a general finite graph. It was known in [Bianchi, Dommers and Giardin`a, EJP 2017] that the model exhibits a metastable behavior in the limit of vanishing diffusivity, and an hierarchical structure of metastability was predicted to occur. Following the partial result in [K., PTRF 2021] which focused on the second time scale of metastable transitions, in this talk we investigate the third time scale, which is indeed the last one. The metastable analysis of the third time scale turns out to be much more complicated than in the previous two situations, mainly due to the fact that the energy landscape regarding the third time scale is essentially three-dimensional.

We consider Wick ordered solutions to the planar stochastic heat equation, corresponding to a Skorokhod interpretation in the Duhamel integral representation of the equation. We prove that the fluctuations far from the center are given by the one-dimensional stochastic heat equation. This talk is based on a joint work with Jeremy Quastel and Balint Virag.

In the study of singular $SPDEs$, it has been a challenging problem to obtain a simple proof of a general probabilistic convergence result ($BPHZ$ theorem). Differently from Chandra and Hairer’s Feynman diagram approach, Linares, Otto, Tempelmayr, and Tsatsoulis recently proposed an inductive proof based on the spectral gap inequality by using their multiindex language. Inspired by their approach, Hairer and Steele also obtained an inductive proof by using the regularity structure language. In this talk, we introduce an extension of the regularity structure including integrability exponents, and provide a simpler proof of $BPHZ$ theorem. This talk is based on a joint work with Ismael Bailleul(Universit ́e de Bretagne Occidentale).

We discuss scaling limits for the Glauber-Kawasaki process. The Glauber-Kawasaki process has been introduced by De Masi et al. to study a reaction-diffusion equation from a microscopic interacting system. They have derived a reaction-diffusion equation as a limiting equation of the density of particles. This limit is usually called the hydrodynamic limit. In this talk, I will focus on several scaling limits related to this hydrodynamic limit. Especially, I will discuss a sharp interface limit for this particle system and its large deviation rate function. The main part of this talk is based on a joint work with Takashi Kagaya. (arXiv:2402.12155).

We study the one-dimensional symmetric simple exclusion process (SSEP). Initially, we put one particle at the origin, called the tagged particle, and put a particle at every other site with probability rho. We proved moderate deviation principles (MPD) for the sample path of the tagged particle. The proof uses MPD from hydrodynamic limits of the SSEP, as proved by Gao and Quastel (2003). The talk is based on joint work with Xiaofeng Xue.

We consider the supercritical bond percolation on the $\mathbb{Z}^{d}$ lattice and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known from the Kingman subadditive ergodic theorem that there exists a deterministic constant $μ(x)$ such that the chemical distance $D(0, nx)$ between two connected points 0 and $nx$ grows like $nμ(x)$. Garet and Marchand prove that the probability of the upper tail large deviation event ${D(0, nx)>μ(nx)(1 +ε),0↔nx}$ decays exponentially with $n$. In this talk, we discuss the existence of the rate function for the upper tail large deviation when $d ≥3$ and $ε >0$ is small enough. Moreover, for $d ≥3$, we prove that the upper tail large deviation event is created by space-time cut-points (points that any geodesic from 0 to $nx$ must cross at a given time) that forces the geodesics to go in a non-optimal direction or to wiggle significantly before reaching the cut-point, where the geodesics extra time. This enables us to express the rate function in terms of the rate function for a space-time cut-point. This talk is based on joint work with Barbara Dembin ($ETH$).

Functional inequalities provide us very robust and powerful tools to analyze the problem of convergence to equilibrium. Perhaps the most famous example is the application of functional inequalities to the overdamped Langevin equation (or the Fokker-Planck equation): Poincar\'e inequalities for the invariant measure imply exponential decay in variance, while logarithmic Sobolev inequalities imply exponential decay in entropy, and both yield explicit rates of convergence. These results give strong motivation for establishing functional inequalities for various kinetic models. In this talk, I'll speak about Poincar\'e and log Sobolev inequalities for a mean-field particle system which is corresponding to the McKean-Vlasov equation. The constants in our estimates are uniform in the number of particles. Moreover, the uniformity in logarithmic Sobolev inequalities enables us to deduce the exponential convergence in free energy of the McKean–Vlasov equation with an explicit rate, either by means of an entropy-entropy production inequality, or by the propagation of chaos property. This talk is based on a joint work with Arnaud Guillin, Wei Liu and Liming Wu.

We investigate the Langevin dynamics of various lattice formulations of the Yang--Mills--Higgs model, where the Higgs component takes values in $\mathbb{R}^N$, $\mathbb{S}^{N-1}$ or a Lie group. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. As an application, we establish that correlations for a broad range of observables decay exponentially. Specifically, the infinite volume measure exhibits a strictly positive mass gap under strong coupling conditions. Moreover, appropriately rescaled observables exhibit factorized correlations in the large $N$ limit when the state space is compact. Our approach involves disintegration and a nuanced analysis of correlations to effectively control the unbounded Higgs component.

In this talk we first recall the classical variational framework for SPDE and briefly review some recent progress in this field. Then we present our results concerning the well-posedness and asymptotics of a class of McKean-Vlasov SPDEs and related stochastic systems.

In many complex stochastic systems exhibiting metastable behavior, e.g., condensing zero-range process, condensing inclusion process, low-temperature Ising model, or Langevin dynamics with complicated potential, it is important to describe the macroscopic behavior of the system by the convergence to a certain limiting Markov chain whose state set consists of the metastable states of the original dynamics. We explain this agenda in more detail, and then review recent development known as the resolvent approach regarding this sort of problems. This is joint work with Claudio Landim, Jungkyoung Lee, and Diego Marcondes.

Motivated by the probabilistic representation of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on flow distribution. We establish the existence and uniqueness of both strong and weak solutions under one-sided Lipschitz conditions and singular drifts. These newly proposed flow-distribution dependent stochastic differential equations are closely connected to quasilinear backward Kolmogorov equations and forward Fokker-Planck equations. Furthermore, we investigate a stochastic version of the 2D-Navier-Stokes equation with fractional Brownian noise. We demonstrate the global well-posedness and smoothness of solutions when the Hurst parameter $H$ lies in the range $(0, \frac12]$ and the initial vorticity is a finite signed measure.

The Mott variable-range hopping is a random walk on a homogeneous Poisson point process whose jump rate to another point depends on the distance to that point. Unlike the multi-dimensional model which usually exhibits the diffusive behavior, sub-diffusive regimes naturally appear in the one-dimensional model. We studied two different sub-diffusive regimes for the one-dimensional Mott variable-range hopping. In the first case, the limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to large gaps in the Poisson point process, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. In the second case, the barriers have stronger effect and we establish that, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Based on joint works with David Croydon and Stefan Junk.

I am going to make some elementary discussions on the presence of (the Euclidean norm of) four-dimensional Brownian motion in the continuum Derrida—Retaux system. Joint work with E. Aïdékon, B. Derrida, T. Duquesne.