We present a new approach, inspired by Stein’s method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions.

The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.

We consider the $N$-particle noncolliding Bernoulli random walk—a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the “bulk”), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk.

The noncolliding Bernoulli random walk is a discrete analogue of the $\boldsymbol{\beta}=2$ Dyson Brownian motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.

Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay between the statistical modeling of network data, the theory of large graph limits, exchangeability and network sampling. The purpose of the present paper is to clarify the relationships between these subjects by explaining each in terms of a certain natural sampling scheme associated with the graphex model. The first main technical contribution is the introduction of sampling convergence, a new notion of graph limit that generalizes left convergence so that it becomes meaningful for the sparse graph regime. The second main technical contribution is the demonstration that the (somewhat cryptic) notion of exchangeability underpinning the graphex framework is equivalent to a more natural probabilistic invariance expressed in terms of the sampling scheme.

We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product $T_{k}\times\mathbb{Z}^{d}$ of a $k$-regular tree ($k\geq3$) with $\mathbb{Z}^{d}$, for which these results were previously only known for large $k$.

In this paper, we study the transition densities of pure-jump symmetric Markov processes in $\mathbb{R}^{d}$, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of the transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at infinity.

As Dembo (In Lectures on Probability Theory and Statistics (2005) 1–101 Springer, and International Congress of Mathematicians, Vol. III (2006) 535–558, Eur. Math. Soc.) suggested, we consider the problem of late points for a simple random walk in two dimensions. It has been shown that the exponents for the number of pairs of late points coincide with those of favorite points and high points in the Gaussian free field, whose exact values are known. We determine the exponents for the number of $j$-tuples of late points on average.

For a given quasi-regular positivity preserving coercive form, we construct a family of ($\sigma$-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.

For a contraction $C_{0}$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincaré inequalities for the symmetric and antisymmetric part of the generator. As applications, nonexponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.

Let $(Z_{t})_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, that is, the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke (Ann. Probab. 15 (1987) 1052–1061) says that this martingale converges almost surely to a limit $Z_{\infty }$, positive on the event of survival. In this paper our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014) 042143): \begin{equation*}\sqrt{t}\bigg(Z_{\infty }-Z_{t}+\frac{\log t}{\sqrt{2\pi t}}Z_{\infty }\bigg)\xrightarrow[t\to \infty ]{}S_{Z_{\infty }}\quad\text{in law},\end{equation*} where $S$ is a spectrally positive 1-stable Lévy process independent of $Z_{\infty }$.

In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_{\infty }$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale.

All proofs in this paper are given under the moment assumption $\mathbb{E}[L(\log L)^{3}]<\infty $, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.

We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. Ann. Probab. 29 (2001) 724–765). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni (Israel J. Math. 148 (2005) 87–113). The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.

As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48 (1987) 1–141) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.

In this short note we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d Gaussian Free Fiedl (GFF) on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method, which we believe is universal in the sense that it can be generalized to all dimensions and to all log-correlated fields. Second, in the 2d case we consider, the value of the constant in front of the inverse power is (up to explicit terms) nothing but the Liouville reflection coefficient taken at a special value. The explicit computation of the constant was performed in the recent rigorous derivation with A. Kupiainen of the DOZZ formula (Kupiainen, Rhodes and Vargas (2017a, 2017b)); to our knowledge, it is the first time one derives rigorously an explicit value for such a constant in the tail expansion of a GMC measure. We have deliberately kept this paper short to emphasize the method so that it becomes an easily accessible toolbox for computing tails in GMC theory.

We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case (Ann. Probab. 22 (1994) 995–1025). Then we establish the maximal weak-type $(1,1)$ inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty $ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-$\lambda$ inequalities.

We show that the convex hull of the path of Brownian motion in $n$-dimensions, up to time $1$, is a smooth set. As a consequence we conclude that a Brownian motion in any dimension almost surely has no cone points for any cone whose dual cone is nontrivial.

We study the mixing time of the unit-rate zero-range process on the complete graph, in the regime where the number $n$ of sites tends to infinity while the density of particles per site stabilizes to some limit $\rho >0$. We prove that the worst-case total-variation distance to equilibrium drops abruptly from $1$ to $0$ at time $n(\rho +\frac{1}{2}\rho^{2})$. More generally, we determine the mixing time from an arbitrary initial configuration. The answer turns out to depend on the largest initial heights in a remarkably explicit way. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase. As time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof combines metastability, separation of timescales, fluid limits, propagation of chaos, entropy and a spectral estimate by Morris (Ann. Probab. 34 (2006) 1645–1664).

We consider random polynomials of the form $H_{n}(z)=\sum_{j=0}^{n}\xi_{j}q_{j}(z)$ where the $\{\xi_{j}\}$ are i.i.d. nondegenerate complex random variables, and the $\{q_{j}(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau $ satisfying the Bernstein–Markov property on a regular compact set $K\subset\mathbb{C}$. We show that if $\mathbb{P}(|\xi_{0}|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_{n}$ converges weakly in probability to the equilibrium measure of $K$. This is the best possible result, in the sense that the roots of $G_{n}(z)=\sum_{j=0}^{n}\xi_{j}z^{j}$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $\xi_{j}$ is not satisfied.

We also consider random polynomials of the form $\sum_{k=0}^{n}\xi_{k}f_{n,k}z^{k}$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{\xi_{k}\}$ satisfy $\mathbb{E}\log (1+|\xi_{0}|)<\infty $. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.

In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Itô, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor (Illinois J. Math. 50 (2006) 413–437), enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein’s theory of strings by showing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, an explicit hypocoercivity $L^{2}$-rate of convergence to equilibrium which is expressed as the spectral gap perturbed by the spectral projection norms.

We prove a local law for the adjacency matrix of the Erdős–Rényi graph $G(N,p)$ in the supercritical regime $pN\geq C\log N$ where $G(N,p)$ has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from (Ann. Probab. 41 (2013) 2279–2375) by extending them all the way down to the critical scale $pN=O(\log N)$.

A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed $\ell^{2}$ and $\ell^{\infty }$ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale $pN=O(\log N)$. These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.

In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4}n\log n-\frac{1}{4}n\log\log{n}$ with window of order $n$, answering a conjecture of Diaconis.

The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient $\rho >0$ and both the population dimension $p$ and the sample size $n$ tend to infinity with $\log p=o(n^{\frac{1}{3}})$. As $0<\rho <1$, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as $0<\rho <1/2$. This differs substantially from a well-known result for the independent case where $\rho =0$, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of $\rho $ where the transition occurs. If $\rho $ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen–Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^{n}$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin–Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin–Wagner theorem applies even though the symmetry groups of $\mathbb{H}^{n}$ and $\mathbb{H}^{2|2}$ are nonamenable.