We study local correlations of certain interacting particle systems on the real line which show repulsion similar to eigenvalues of random Hermitian matrices. Although the new particle system does not seem to have a natural spectral or determinantal representation, the local correlations in the bulk coincide in the limit of infinitely many particles with those known from random Hermitian matrices; in particular they can be expressed as determinants of the so-called sine kernel. These results may provide an explanation for the appearance of sine kernel correlation statistics in a number of situations which do not have an obvious interpretation in terms of random matrices.

The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the singularity of their transition probabilities, while topological and functional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the investigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimensional or weak-∗ ergodic processes. To this end, we first develop local counterparts of zero–two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that admit asymptotic couplings or that are locally mixing in the sense of H. Föllmer, and to a non-Markovian ergodic theorem for stationary absolutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. Finally, we show that our abstract results can be applied to infinite-dimensional Markov processes that arise in several settings, including dissipative stochastic partial differential equations, stochastic spin systems and stochastic differential delay equations.

We prove duality relations for two interacting particle systems: the $q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and the asymmetric simple exclusion process (ASEP). Expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents, respectively. Duality implies that they solve systems of ODEs. These systems are integrable and for particular step and half-stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems’ solutions, and hence also the moments.

We form Laplace transform-like generating functions of these moments and via residue calculus we compute two different types of Fredholm determinant formulas for such generating functions. For ASEP, the first type of formula is new and readily lends itself to asymptotic analysis (as necessary to reprove GUE Tracy–Widom distribution fluctuations for ASEP), while the second type of formula is recognizable as closely related to Tracy and Widom’s ASEP formula [Comm. Math. Phys. 279 (2008) 815–844, J. Stat. Phys. 132 (2008) 291–300, Comm. Math. Phys. 290 (2009) 129–154, J. Stat. Phys. 140 (2010) 619–634]. For $q$-TASEP, both formulas coincide with those computed via Borodin and Corwin’s Macdonald processes [Probab. Theory Related Fields (2014) 158 225–400].

Both $q$-TASEP and ASEP have limit transitions to the free energy of the continuum directed polymer, the logarithm of the solution of the stochastic heat equation or the Hopf–Cole solution to the Kardar–Parisi–Zhang equation. Thus, $q$-TASEP and ASEP are integrable discretizations of these continuum objects; the systems of ODEs associated to their dualities are deformed discrete quantum delta Bose gases; and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics.

We establish that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting “free energy density” which coincides with the (nonrigorous) Bethe prediction of statistical physics. Using this result, we characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs without the use of the second moment method employed in previous works on these questions. As a consequence, we show that for both the hard-core model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on $d$-regular graphs when the model has nonuniqueness on the $d$-regular tree. Together with results of Jerrum–Sinclair, Weitz, and Sinclair–Srivastava–Thurley, this gives an almost complete classification of the computational complexity of homogeneous two-spin systems on bounded-degree graphs.

Random walks in random scenery are processes defined by $Z_{n}:=\sum_{k=1}^{n}\xi_{X_{1}+\cdots+X_{k}}$, where basically $(X_{k},k\ge1)$ and $(\xi_{y},y\in\mathbb{Z})$ are two independent sequences of i.i.d. random variables. We assume here that $X_{1}$ is $\mathbb{Z}$-valued, centered and with finite moments of all orders. We also assume that $\xi_{0}$ is $\mathbb{Z}$-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that $(n^{-3/4}Z_{[nt]},t\ge0)$ converges in distribution as $n\to\infty$ toward some self-similar process $(\Delta_{t},t\ge0)$ called Brownian motion in random scenery. In a previous paper, we established that $\mathbb{P}(Z_{n}=0)$ behaves asymptotically like a constant times $n^{-3/4}$, as $n\to\infty$. We extend here this local limit theorem: we give a precise asymptotic result for the probability for $Z$ to return to zero simultaneously at several times. As a byproduct of our computations, we show that $\Delta$ admits a bi-continuous version of its local time process which is locally Hölder continuous of order $1/4-\delta$ and $1/6-\delta$, respectively, in the time and space variables, for any $\delta>0$. In particular, this gives a new proof of the fact, previously obtained by Khoshnevisan, that the level sets of $\Delta$ have Hausdorff dimension a.s. equal to $1/4$. We also get the convergence of every moment of the normalized local time of $Z$ toward its continuous counterpart.

We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb{P} (|X_{ij}|\geq t)$ behave like $e^{-at^{\alpha }}$ for some $a>0$ and $\alpha \in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+\alpha /2}$ with a good rate function $J(\mu)$ that is finite only if $\mu$ is of the form $\mu=\mu_{\mathrm{sc}}\boxplus\nu$ for some probability measure $\nu$ on $\mathbb{R} $, where $\boxplus$ denotes the free convolution and $\mu_{\mathrm{sc}}$ is Wigner’s semicircle law. We obtain explicit expressions for $J(\mu_{\mathrm{sc}}\boxplus\nu)$ in terms of the $\alpha $th moment of $\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.

Consider a sequence of possibly random graphs $G_{N}=(V_{N},E_{N})$, $N\ge1$, whose vertices’s have i.i.d. weights $\{W^{N}_{x}:x\in V_{N}\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^{N}_{t}$, $t\ge0$, be a continuous time simple random walk on $G_{N}$ which waits a mean $W^{N}_{x}$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge2$, random $d$-regular graphs and the largest component of super-critical Erdős–Rényi random graphs.

We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite $p$th moment, for every $p<d$, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.

Motivated by the study of existence, uniqueness and regularity of solutions to stochastic partial differential equations driven by jump noise, we prove Itô isomorphisms for $L^{p}$-valued stochastic integrals with respect to a compensated Poisson random measure. The principal ingredients for the proof are novel Rosenthal type inequalities for independent random variables taking values in a (noncommutative) $L^{p}$-space, which may be of independent interest. As a by-product of our proof, we observe some moment estimates for the operator norm of a sum of independent random matrices.