We explain the large deviation behavior of the totally asymmetric simple exclusion process in one dimension.

We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the equidecay profiles. Finally, we explain the relation between these results from Statistical Mechanics and their counterparts in Quantum Field Theory.

Following an idea introduced by Carlen, Carvalho and Loss [7] we propose a general strategy to prove Poincaré inequalities in product spaces with one or more conservation laws. The method is shown to yield alternative proofs of well known results, such as the diffusive bounds for the spectral gap of generalized exclusion and zero range processes. Other models are also discussed, including anisotropic exclusion processes, simple exclusion with site–disorder and Ginzburg–Landau processes, where this approach provides sharp spectral gap estimates apparently inaccessible by previously known techniques.

The traditional ergodicity consists a crucial part in the theory of stochastic processes, plays a key role in practical applications. The ergodicity has much refined recently, due to the study on some inequalities, which are especially powerful in the infinite dimensional situation. The explicit criteria for various types of ergodicity for birth-death processes and one-dimensional diffusions are collected in Tables 8.1 and 8.2, respectively. In particular, an interesting story about how to obtain one of the criteria for birth-death processes is explained in details. Besides, a diagram for various types of ergodicity for general reversible Markov processes is presented.

Directed polymers in random environment can be thought of as a model of statistical mechanics in which paths of stochastic processes interact with a quenched disorder (impurities), depending on both time and space. We review here main results which have been obtained during the last fifteen years, with proofs to most of the results. The material covers the diffusive behavior of the polymers in weak disorder phase studied by J. Imbrie, T. Spencer, E. Bolthausen, R. Song and X. Y. Zhou [11, 3, 25], and localization of the paths in strong disordered phase recently obtained by P. Carmona, Y. Hu, and the authors of the present article [4, 5].

The hyperbolic (Euler) scaling limit of weakly asymmetric Ginzburg–Landau models with a single conservation law is investigated, weak asymmetry means that the microscopic viscosity of the system tends to infinity in a prescribed way during the hydrodynamic limit. The system is not attractive, its potential is a bounded perturbation of a quadratic function. The macroscopic equation reads as $\partial_t \rho + \partial_x S'(\rho) = 0$, where $S$ is a convex function. The Tartar - Murat theory of compensated compactness is extended to microscopic systems, we prove weak convergence of the scaled density field to the set of weak solutions. In the attractive case of a convex potential this set consists of the unique entropy solution. Our main tool is the logarithmic Sobolev inequality of Landim, Panizo and Yau for continuous spins.

We consider the $\nabla_{\varphi}$ interface model with weak self potential (one-body potential) under general Dirichlet boundary conditions on a large bounded domain and establish the large deviation principle for the macroscopically scaled interface height variables. As its application the law of large numbers is proved and the limit profile is characterized by a variational problem which was studied by Alt-Caffarelli [1], Alt-Caffarelli-Friedman [2] and others. The minimizers generate free boundaries inside the domain. We also discuss the $\nabla_{\varphi}$ interface model with $\delta$-pinning potential in one dimension.

We consider the fluctuation of the phase boundary separating two phases of the Widom-Rowlinson model in the plane square lattice. The phase boundary is conditioned to have specified values of the area underneath and the height difference of two end points. Dobrushin and Hryniv studied the phase boundary of the Solid-on-Solid model [DH1] and of the Ising model [DH2], and obtained the central limit theorem for the fluctuation of the phase boundary from the Wulff profile. The phase boundary of the Ising model is well approximated by that of the Solid-on-Solid model with the aid of the cluster expansion. Their argument seems to be applicable to the general models which have polymer representation. We apply their theory to the Widom-Rowlinson model.

Non-colliding Brownian particles in one dimension is studied. $N$ Brownian particles start from the origin at time 0 and then they do not collide with each other until finite time $T$. We derive the determinantal expressions for the multitime correlation functions using the self-dual quaternion matrices. We consider the scaling limit of the infinite particles $N \to \infty$ and the infinite time interval $T \to \infty$. Depending on the scaling, two limit theorems are proved for the multitime correlation functions, which may define temporally inhomogeneous infinite particle systems.

We consider the asymmetric simple exclusion process in dimension $d \ge 3$. We review some results concerning the equilibrium bulk fluctuations and the asymptotic behaviour of a second class particle.

Dyson's model on interacting Brownian particles is a stochastic dynamics consisting of an infinite amount of particles moving in $\mathbb{R}$ with a logarithmic pair interaction potential. For this model we will prove that each pair of particles never collide.

The equilibrium state of this dynamics is a determinantal random point field with the sine kernel. We prove for stochastic dynamics given by Dirichlet forms with determinantal random point fields as equilibrium states the particles never collide if the kernel of determining random point fields are locally Lipschitz continuous, and give examples of collision when Hölder continuous.

In addition we construct infinite volume dynamics (a kind of infinite dimensional diffusions) whose equilibrium states are determinantal random point fields. The last result is partial in the sense that we simply construct a diffusion associated with the maximal closable part of canonical pre Dirichlet forms for given determinantal random point fields as equilibrium states. To prove the closability of canonical pre Dirichlet forms for given determinantal random point fields is still an open problem. We prove these dynamics are the strong resolvent limit of finite volume dynamics.

We show that the grand canonical ensembles of ideal gas under Fermi, Boson and other statistics give simple examples of the random point fields studied in the previous papers [13, 14, 15]. Also we present two classes of nonsymmetric integral operators for which such random point fields do exist.

Some general properties of Lévy processes conditioned to stay positive are studied. As an application, which is our main concern, a result of localization is obtained for diffusion processes in Lévy environments.