We prove a color-position symmetry for a class of ASEP-like interacting particle systems with discrete time on the one-dimensional lattice. The full space-time inhomogeneity of our systems allows to apply the result to colored (or multi-species) ASEP and stochastic vertex models for a certain class of initial/boundary conditions, generalizing previous results of Amir–Angel–Valko and Borodin–Wheeler. We are also able to use the symmetry, together with previously known results for uncolored models, to find novel asymptotic behavior of the second-class particles in several situations.

Let M be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the M-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on M run for m steps is of order ${\mathit{m}}^{2/\mathit{d}+{\mathit{o}}_{\mathit{m}}(1)}$, where d is the metric ball volume growth exponent for M (which was shown to exist by Ding and Gwynne (Comm. Math. Phys. 374 (2020) 1877–1934). By known bounds for d, one has $0.55051\cdots \le 2/\mathit{d}\le 0.563315\dots$.

Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance ${\mathit{m}}^{2/\mathit{d}+{\mathit{o}}_{\mathit{m}}(1)}$ in m units of time and that the graph-distance diameter of a finite spanning-tree-weighted random planar map with n edges, with or without boundary, is of order ${\mathit{n}}^{1/\mathit{d}+{\mathit{o}}_{\mathit{n}}(1)}$ except on an event with probability decaying faster than any negative power of n.

Our proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity.

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to γ-Liouville quantum gravity (γ-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and γ ranges from 0 to 2 as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling ${\mathrm{SLE}}_{\mathit{\kappa }}$ for $\mathit{\kappa }=16/{\mathit{\gamma }}^2$ (in the annealed sense) and that simple random walk on the embedded map converges to Brownian motion (in the quenched sense).

This work constitutes the first proof that a discrete conformal embedding of a random planar map converges to LQG. Many more such statements have been conjectured. Since the mated-CRT map can be viewed as a coarse-grained approximation to other random planar maps (the UIPT, tree-weighted maps, bipolar-oriented maps, etc.), our results indicate a potential approach for proving that embeddings of these maps converge to LQG as well.

To prove the main result, we establish several (independently interesting) theorems about LQG surfaces decorated by space-filling SLE. There is a natural way to use the SLE curve to divide the plane into “cells” corresponding to vertices of the mated-CRT map. We study the law of the shape of the origin-containing cell, in particular proving moments for the ratio of its squared diameter to its area. We also give bounds on the degree of the origin-containing cell and establish a form of ergodicity for the entire configuration. Ultimately, we use these properties to show (with the help of a general theorem proved in a separate paper) that random walk on these cells converges to a time change of Brownian motion, which in turn leads to the Tutte embedding result.

We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion B on every compact interval. In particular, the Airy${}_1$ process is absolutely continuous with respect to B on any compact interval.

The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools, which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for nonintersecting Brownian ensembles, and a quick proof of tightness for Dyson’s Brownian motion converging to the Airy line ensemble.

We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Lévy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically nonexplicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.

In this paper, we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erdős–Rényi random graph ${\mathcal{G}}_{\mathit{n},\mathit{p}}$, in the regime of p where the edge of the spectrum is no longer governed by global observables, such as the number of edges, but rather by localized statistics, such as high degree vertices. Going beyond the recent developments in mean-field approximations of related problems, this paper provides a comprehensive treatment of the large deviations of the spectral edge in this entire regime, which notably includes the well-studied case of constant average degree. In particular, for $\mathit{r}\ge 1$ fixed, we pin down the asymptotic probability that the top r eigenvalues are jointly greater/less than their typical values by multiplicative factors bigger/smaller than 1, in the regime mentioned above. The proof for the upper tail relies on a novel structure theorem, obtained by building on estimates in (Combin. Probab. Comput. 12 (2003) 61–72), followed by an iterative cycle removal process, which shows, conditional on the upper tail large deviation event, with high probability the graph admits a decomposition in to a disjoint union of stars and a spectrally negligible part. On the other hand, the key ingredient in the proof of the lower tail is a Ramsey-type result which shows that if the Kth largest degree of a graph is not atypically small (for some large K depending on r), then either the top eigenvalue or the rth largest eigenvalue is larger than that allowed by the lower tail event on the top r eigenvalues, thus forcing a contradiction. The above arguments reduce the problems to developing a large deviation theory for the extremal degrees which could be of independent interest.

We consider words ${\mathit{G}}_{{\mathit{i}}_1}\cdots {\mathit{G}}_{{\mathit{i}}_{\mathit{m}}}$ involving i.i.d. complex Ginibre matrices and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length m is a Fuss–Catalan distribution with parameter $\mathit{m}+1$. This generalizes previous results concerning powers of a complex Ginibre matrix and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, band matrices, and sparse matrices.

These results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically.

In this paper we consider the Anderson Hamiltonian with white noise potential on the box ${[0,\mathit{L}]}^2$ with Dirichlet boundary conditions. We show that all of the eigenvalues divided by $log\mathit{L}$, converge as $\mathit{L}\to \infty$, almost surely to the same deterministic constant which is given by a variational formula.

In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular, a family of models which, for any $\mathit{k}\in \mathbb{N}$, includes models with k smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a nonintersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel, we obtain the local behavior in the smooth and rough regions, as the size of the Aztec diamond tends to infinity. From the mentioned limit the macroscopic picture, such as the arctic curves and, in particular, the number of smooth regions, is deduced. Moreover, we compute the limit of the height function, and, as a consequence, we confirm in the setting of this paper that the limit in the rough region fulfills the complex Burgers’ equation, as stated by Kenyon and Okounkov.

We confirm the long-standing prediction that $\mathit{c}=\mathit{e}\approx 2.718$ is the threshold for the emergence of a nonvanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős–Rényi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent ${(\mathit{A}-\mathit{z})}^{-1}$ near the singular point $\mathit{z}=0$, where A is the adjacency operator of the Poisson–Galton–Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton–Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.

We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős–Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measure-valued differential equation. The main technique is to observe that, conditioned on the vertex ages, the graph is an inhomogeneous random graph in the sense of Bollobás, Janson and Riordan. We then study the evolution of the ages via the multitype Galton–Watson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.

Consider the Ising model at low temperatures and positive external field λ on an $\mathit{N}\times \mathit{N}$ box with Dobrushin boundary conditions that are plus on the north, east and west boundaries and minus on the south boundary. If $\mathit{\lambda }=0$, the interface separating the plus and minus phases is diffusive, having $\mathit{O}(\sqrt{\mathit{N}})$ height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are $\mathit{O}(1)$, and the interface is only partially wetted, being pinned to its southern boundary. We study the critical prewetting regime of ${\mathit{\lambda }}_{\mathit{N}}\downarrow 0$, where the height fluctuations are expected to scale as ${\mathit{\lambda }}^{-1/3}$ and the rescaled interface is predicted to converge to the Ferrari–Spohn diffusion. Velenik (Probab. Theory Related Fields 129 (2004) 83–112) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts.

In this paper we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical prewetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving ${\mathit{e}}^{-\mathrm{\Theta }({\mathit{x}}^{3/2})}$ upper tails reminiscent of the Tracy–Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines and the polylogarithmic prefactor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface with novel local resampling and coupling schemes.