We study the persistence probability of a centered stationary Gaussian process on $\mathbb{Z}$ or $\mathbb{R}$, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a “mating-of-trees” type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the γ-mated-CRT map for $\mathit{\gamma }\in (0,2)$. For each of these maps, we obtain an upper bound for the Green’s function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after n steps, and a lower bound for the graph-distance displacement of the random walk, all of which are sharp up to polylogarithmic factors.

When combined with work of Lee (2017), our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s. equal to 2, that is, the (quenched) probability that the simple random walk returns to its starting point after $2\mathit{n}$ steps is ${\mathit{n}}^{-1+{\mathit{o}}_{\mathit{n}}(1)}$. Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least ${\mathit{n}}^{1/4-{\mathit{o}}_{\mathit{n}}(1)}$ units of graph distance in n units of time. The matching upper bound for the displacement is proven by Gwynne and Hutchcroft (Probab. Theory Related Fields178 (2020) 567–611). These two works together resolve a conjecture of Benjamini and Curien (Geom. Funct. Anal.23 (2013) 501–531) in the UIPT case.

Our proofs are based on estimates for the mated-CRT map (which come from its relationship to SLE-decorated Liouville quantum gravity) and a strong coupling of the mated-CRT map with the other random planar map models.

We analyze the local and global smoothing rates of the smoothing process and obtain convergence rates to stationarity for the dual process known as the potlatch process. For general finite graphs we connect the smoothing and convergence rates to the spectral gap of the associated Markov chain. We perform a more detailed analysis of these processes on the torus. Polynomial corrections to the smoothing rates are obtained. They show that local smoothing happens faster than global smoothing. These polynomial rates translate to rates of convergence to stationarity in ${\mathit{L}}^2$-Wasserstein distance for the potlatch process on ${\mathbb{Z}}^{\mathit{d}}$.

We work under the Aïdékon–Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that $\mathbb{E}\mathit{Z}{\mathbb{1}}_{\{\mathit{Z}\le \mathit{x}\}}=log\mathit{x}+\mathit{o}(log\mathit{x})$ as $\mathit{x}\to \infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E}\mathit{Z}{\mathbb{1}}_{\{\mathit{Z}\le \mathit{x}\}}=log\mathit{x}+\mathrm{const}+\mathit{o}(1)$ as $\mathit{x}\to \infty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit Z. The methodological novelty of the present paper is a three terms representation of a subharmonic function of, at most, linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.

We consider a Poisson equation in ${\mathbb{R}}^{\mathit{d}}$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multitime scales stochastic differential equations with only Hölder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with respect to the slow variable and does not depend on their regularities with respect to the fast component.

We study a particular model of a random medium, called the orthant model, in general dimensions $\mathit{d}\ge 2$. Each site $\mathit{x}\in {\mathbb{Z}}^{\mathit{d}}$ independently has arrows pointing to its positive neighbours $\mathit{x}+{\mathit{e}}_{\mathit{i}}$, $\mathit{i}=1,\dots ,\mathit{d}$ with probability p and, otherwise, to its negative neighbours $\mathit{x}-{\mathit{e}}_{\mathit{i}}$, $\mathit{i}=1,\dots ,\mathit{d}$ (with probability $1-\mathit{p}$). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when p is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary which is a key requirement of the subadditive ergodic theorem.

We consider exit problems for small, white noise perturbations of a dynamical system generated by a vector field and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Furthermore, we compute the limiting exit distributions conditioned on atypical exit events of polynomially small probability and show that the limits are Radon–Nikodym equivalent to volume measures on certain manifolds that we construct. This situation is in sharp contrast with the large deviation picture where the limiting conditional distributions are point masses.

We show that, for an $\mathit{n}\times \mathit{n}$ random matrix A with independent uniformly anticoncentrated entries such that $\mathbb{E}\Vert \mathit{A}{\Vert }_{\mathrm{HS}}^2\le \mathit{K}{\mathit{n}}^2$, the smallest singular value ${\mathit{\sigma }}_{\mathit{n}}(\mathit{A})$ of A satisfies

$${\mathbb{P}}_{}\{{\mathit{\sigma }}_{\mathit{n}}(\mathit{A})\le \frac{\mathit{\epsilon }}{\sqrt{\mathit{n}}}\}\le \mathit{C}\mathit{\epsilon }+2{\mathit{e}}^{-\mathit{c}\mathit{n}},\mathit{\epsilon }\ge 0.$$

This extends earlier results (Adv. Math.218 (2008) 600–633; Israel J. Math.227 (2018) 507–544) by removing the assumption of mean zero and identical distribution of the entries across the matrix as well as the recent result (Livshyts (2018)) where the matrix was required to have i.i.d. rows. Our model covers inhomogeneous matrices allowing different variances of the entries as long as the sum of the second moments is of order $\mathit{O}({\mathit{n}}^2)$.

In the past advances, the assumption of i.i.d. rows was required due to lack of Littlewood–Offord-type inequalities for weighted sums of non-i.i.d. random variables. Here, we overcome this problem by introducing the Randomized Least Common Denominator (RLCD) which allows to study anti-concentration properties of weighted sums of independent but not identically distributed variables. We construct efficient nets on the sphere with lattice structure and show that the lattice points typically have large RLCD. This allows us to derive strong anticoncentration properties for the distance between a fixed column of A and the linear span of the remaining columns and prove the main result.

We show uniqueness in law for the critical SPDE

$$\mathit{d}{\mathit{X}}_{\mathit{t}}=\mathit{A}{\mathit{X}}_{\mathit{t}}\mathit{d}\mathit{t}+{(-\mathit{A})}^{1/2}\mathit{F}(\mathit{X}(\mathit{t}))\mathit{d}\mathit{t}+\mathit{d}{\mathit{W}}_{\mathit{t}},{\mathit{X}}_0=\mathit{x}\in \mathit{H},$$

where $\mathit{A}:\text{dom}(\mathit{A})\subset \mathit{H}\to \mathit{H}$ is a negative definite self-adjoint operator on a separable Hilbert space H having ${\mathit{A}}^{-1}$ of trace class and W is a cylindrical Wiener process on H. Here, $\mathit{F}:\mathit{H}\to \mathit{H}$ can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn–Hilliard-type equations which have interesting applications. To get weak uniqueness, we use an infinite dimensional localization principle and also establish a new optimal regularity result for the Kolmogorov equation $\mathit{\lambda }\mathit{u}-\mathit{L}\mathit{u}=\mathit{f}$ associated to the SPDE when $\mathit{F}=0$ ($\mathit{\lambda }>0$, $\mathit{f}:\mathit{H}\to \mathbb{R}$ Borel and bounded). In particular, we prove that the first derivative $\mathit{D}\mathit{u}(\mathit{x})$ belongs to $\text{dom}({(-\mathit{A})}^{1/2})$, for any $\mathit{x}\in \mathit{H}$, and ${sup}_{\mathit{x}\in \mathit{H}}|{(-\mathit{A})}^{1/2}\mathit{D}\mathit{u}(\mathit{x}){|}_{\mathit{H}}=\Vert {(-\mathit{A})}^{1/2}\mathit{D}\mathit{u}{\Vert }_0\le \mathit{C}\Vert \mathit{f}{\Vert }_0$.

We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdős–Rényi graph $\mathbb{G}(\mathit{N},\mathit{d}/\mathit{N})$ in the critical regime $\mathit{d}\asymp log\mathit{N}$ of the transition uncovered in (Ann. Inst. Henri Poincaré Probab. Stat.56 (2020) 2141–2161; Ann. Probab.47 (2019) 1653–1676), where the regimes $\mathit{d}\gg log\mathit{N}$ and $\mathit{d}\ll log\mathit{N}$ were studied. We establish a one-to-one correspondence between vertices of degree at least $2\mathit{d}$ and nontrivial (excluding the trivial top eigenvalue) eigenvalues of $\mathit{A}/\sqrt{\mathit{d}}$ outside of the asymptotic bulk $[-2,2]$. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value $\mathit{d}={\mathit{d}}_{\ast }=\frac{1}{log4-1}log\mathit{N}$. For $\mathit{d}<{\mathit{d}}_{\ast }$, we obtain rigidity bounds on the locations of all eigenvalues outside the interval $[-2,2]$, and for $\mathit{d}>{\mathit{d}}_{\ast }$, we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities.

Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara–Bass formula (Ann. Inst. Henri Poincaré Probab. Stat.56 (2020) 2141–2161). Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of A and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the ${\ell }^2$-norms of the rows.

We study the two-dimensional wave equation with cubic nonlinearity posed on ${\mathbb{R}}^2$ with space-time white noise forcing. After a suitable renormalisation of the nonlinearity, we prove global well-posedness for this equation for initial data in ${\mathcal{H}}^{\mathit{s}}$, $\mathit{s}>\frac{4}{5}$.

Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 3$, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in ${\mathbb{R}}^{\mathit{d}}$ as an isometry-invariant random partition of ${\mathbb{R}}^{\mathit{d}}$ to bounded polyhedra, and also as an isometry-invariant random partition of ${\mathbb{R}}^{\mathit{d}}$ to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID’s.

We consider additive functionals of stationary Markov processes and show that under Kipnis–Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First, we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a nonreversible Ornstein–Uhlenbeck process, while the last example is a diffusion in a periodic environment.

As a technical step, we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder–Davis–Gundy inequality for local martingale rough paths of (In Séminaire de Probabilités XLI (2008) 421–438 Springer; In Probability and Analysis in Interacting Physical Systems (2019) 17–48 Springer; J. Differential Equations264 (2018) 6226–6301) to the case where only the integrator is a local martingale.

We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $\mathit{\chi }\in (1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of nonextinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on χ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of the graph goes to infinity.

Fix a graph H and some $\mathit{p}\in (0,1)$, and let ${\mathit{X}}_{\mathit{H}}$ be the number of copies of H in a random graph $\mathbb{G}(\mathit{n},\mathit{p})$. Random variables of this form have been intensively studied since the foundational work of Erdős and Rényi. There has been a great deal of progress over the years on the large-scale behaviour of ${\mathit{X}}_{\mathit{H}}$, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that ${\mathit{X}}_{\mathit{H}}$ falls in some small interval or is equal to some particular value? In this paper, we prove the almost-optimal result that if H is connected then for any $\mathit{x}\in \mathbb{N}$ we have $Pr({\mathit{X}}_{\mathit{H}}=\mathit{x})\le {\mathit{n}}^{1-\mathit{v}(\mathit{H})+\mathit{o}(1)}$. Our proof proceeds by iteratively breaking ${\mathit{X}}_{\mathit{H}}$ into different components which fluctuate at “different scales”, and relies on a new anti-concentration inequality for random vectors that behave “almost linearly.”

In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in ${\mathbb{R}}^2$ whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in (Acta Math.205 (2010) 19–104) by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object.

Our main motivation is the study of two models of dynamical critical Voronoi percolation in the plane. In the first model, the Voronoi tiling does not evolve in time while the colors of the cells are resampled at rate 1. In the second model, the centers of the cells move according to (independent) long range stable Lévy processes but the colors do not evolve in time. We prove that for these two dynamical processes there exist almost surely exceptional times with an unbounded monochromatic component.