We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite $d$-regular graphs. It follows that the a finite $d$-regular Ramanujan graph $G$ contains a negligible number of cycles of size less than $c\log\log\vert G\vert$.

We prove that infinite $d$-regular Ramanujan unimodular random graphs are trees. Through Benjamini–Schramm convergence this leads to the following rigidity result. If most eigenvalues of a $d$-regular finite graph $G$ fall in the Alon–Boppana region, then the eigenvalue distribution of $G$ is close to the spectral measure of the $d$-regular tree. In particular, $G$ contains few short cycles.

In contrast, we show that $d$-regular unimodular random graphs with maximal growth are not necessarily trees.

Given a simply connected planar domain $D$, distinct points $x,y\in\partial D$, and $\kappa>0$, the Schramm–Loewner evolution $\operatorname{SLE}_{\kappa}$ is a random continuous non-self-crossing path in $\overline{D}$ from $x$ to $y$. The $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ processes, defined for $\rho_{1},\rho_{2}>-2$, are in some sense the most natural generalizations of $\operatorname{SLE}_{\kappa}$.

When $\kappa\leq4$, we prove that the law of the time-reversal of an $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ from $x$ to $y$ is, up to parameterization, an $\operatorname{SLE}_{\kappa}(\rho_{2};\rho_{1})$ from $y$ to $x$. This assumes that the “force points” used to define $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ are immediately to the left and right of the $\operatorname{SLE}$ seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit $\partial D\setminus\{x,y\}$.

The proof of time-reversal symmetry makes use of the interpretation of $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ as a ray of a random geometry associated to the Gaussian-free field. Within this framework, the time-reversal result allows us to couple two instances of the Gaussian-free field (with different boundary conditions) so that their difference is almost surely constant on either side of the path. In a fairly general sense, adding appropriate constants to the two sides of a ray reverses its orientation.

Moderate deviation principles for stochastic differential equations driven by a Poisson random measure (PRM) in finite and infinite dimensions are obtained. Proofs are based on a variational representation for expected values of positive functionals of a PRM.

We study a new class of time inhomogeneous Pólya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.

We consider a class of kinetically constrained interacting particle systems on $\mathbb{Z}^{d}$ which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of $\mathbb{Z}^{d}$, to each occupation variable $\eta_{x}\in\{0,1\}$ a new value is proposed by tossing a $(1-q)$-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For $d=1$, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On $\mathbb{Z}^{2}$, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit $q\searrow0$ of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on $\mathbb{Z}^{d}$ scales as the $1/d$-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex $x$ out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the $\ell_{1}$-norm of $x$ but also on its direction.

This paper is about the construction of displacement interpolations of probability distributions on a discrete metric graph. Our approach is based on the approximation of any optimal transport problem whose cost function is a distance on a discrete graph by a sequence of entropy minimization problems under marginal constraints, called Schrödinger problems, which are associated with random walks. Displacement interpolations are defined as the limit of the time-marginal flows of the solutions to the Schrödinger problems as the jump frequencies of the random walks tend down to zero. The main convergence results are based on $\Gamma$-convergence of entropy minimization problems.

As a by-product, we obtain new results about optimal transport on graphs.

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^{1}$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014–1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294–4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

We provide a geometric representation of the Poisson and Martin boundaries of a transient, bounded degree triangulation of the plane in terms of its circle packing in the unit disc. (This packing is unique up to Möbius transformations.) More precisely, we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk, and that the space of extremal positive harmonic functions, that is, the Martin boundary, is homeomorphic to the unit circle.

All our results hold more generally for any “good”-embedding of planar graphs, that is, an embedding in the unit disc with straight lines such that angles are bounded away from $0$ and $\pi$ uniformly, and lengths of adjacent edges are comparable. Furthermore, we show that in a good embedding of a planar graph the probability that a random walk exits a disc through a sufficiently wide arc is at least a constant, and that Brownian motion on such graphs takes time of order $r^{2}$ to exit a disc of radius $r$. These answer a question recently posed by Chelkak (2014).

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

Let $\{{X}_{k}\}_{k\geq\mathbb{Z}}$ be a stationary sequence. Given $p\in(2,3]$ moments and a mild weak dependence condition, we show a Berry–Esseen theorem with optimal rate $n^{p/2-1}$. For $p\geq4$, we also show a convergence rate of $n^{1/2}$ in $\mathcal{L}^{q}$-norm, where $q\geq1$. Up to $\log n$ factors, we also obtain nonuniform rates for any $p>2$. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics.

In this paper we prove an approximate continuity result for stochastic differential equations with normal reflections in domains satisfying Saisho’s conditions, which together with the Wong–Zakai approximation result completes the support theorem for such diffusions in the uniform convergence topology. Also by adapting Millet and Sanz-Solé’s idea, we characterize in Hölder norm the support of diffusions reflected in domains satisfying the Lions–Sznitman conditions by proving limit theorems of adapted interpolations. Finally we apply the support theorem to establish a boundary-interior maximum principle for subharmonic functions.

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops (“loop soup”) of parameter $\frac{1}{2}$ associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter $\frac{1}{2}$ do not percolate on periodic lattices. We also construct a coupling between the random interlacement on $\mathbb{Z}^{d}$, $d\geq 3$, introduced by Sznitman, and the Gaussian free field, such that the set of vertices visited by the interlacement is contained in a one-sided level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement.

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein’s method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy–Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale $N^{2}$.

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in the limit of large $N$.