We study critical branching random walks and introduce the branching capacity and related concepts for any finite subset of ℤ^d, d≥ 5 . By introducing these concepts, we can obtain analogues of various classical results for random walks. In this paper, we develop the basic properties of the branching capacity and show that it is closely related to the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point, as well as to the conditional distribution of the hitting point(s).
We study a general class of nonlinear Ginzburg–Landau SPDEs in infinite volume under weak nonlinearity scaling and with non-equilibrium initial data. We derive the KPZ equation as a continuum limit of these equations. This makes rigorous the original derivation of the KPZ equation from physics [31] in the full-space setting, which was a problem posed in [27]. Our analysis is based on a stochastic heat kernel for a linearization of said SPDEs.
We study zero sets of twisted stationary Gaussian random functions on the complex plane, i.e., Gaussian random functions that are stochastically invariant under the action of the Weyl-Heisenberg group. This model includes translation-invariant Gaussian entire functions (GEFs), and also many other non-analytic examples, in which case winding numbers around zeros can be either positive or negative. We investigate zero statistics both when zeros are weighted with their winding numbers (charged zero set) and when they are not (uncharged zero set). We show that the variance of the charged zero statistic always grows linearly with the radius of the observation disk (hyperuniformity). Importantly, this holds for functions with possibly non-zero means and without assuming additional symmetries such as radiality. With respect to uncharged zero statistics, we provide an example for which the variance grows with the area of the observation disk (non-hyperuniformity). This is used to show that, while the zeros of GEFs are hyperuniform, the set of their critical points fails to be so. Our work contributes to recent developments in statistical signal processing, where the time-frequency profile of a non-stationary signal embedded into noise is revealed by performing a statistical test on the zeros of its spectrogram (“silent points”). We show that empirical spectrogram zero counts enjoy moderate deviations from their ensemble averages over large observation windows (something that was previously known only for pure noise). In contrast, we also show that spectrogram maxima (“loud points”) fail to enjoy a similar property. This gives the first formal evidence for the statistical superiority of silent points over the competing feature of loud points, a fact that has been noted by practitioners. In the same vein, our second order asymptotics for spectrogram maxima show that certain heuristic proxy models used in signal processing are inaccurate at large scales.
We consider an infinite system of interacting Brownian motions that preserves a given random point field invariant. Such dynamics are constructed using Dirichlet form theory, which naturally leads to two Dirichlet forms for the random point field: the upper and the lower Dirichlet forms. A fundamental question is the uniqueness of the Dirichlet form: that is, whether these two forms coincide. This uniqueness has often been imposed as a key assumption in the Dirichlet form approach to the stochastic analysis for infinite particle systems. A sufficient condition for the uniqueness of the Dirichlet forms is known when the random point field is tail trivial. However, tail triviality has been established for only a limited class of random point fields. In this paper, we prove the uniqueness of the Dirichlet form without assuming tail triviality. The main contribution of this work is to establish the tail preserving property, which asserts that global properties of the system, such as particle density, are preserved under time evolution. As a consequence, our results also imply the strong uniqueness of solutions to the associated infinite-dimensional stochastic differential equations.
We analyse the aggregate Loewner evolution (ALE), introduced in 2018 by Sola, Turner and Viklund to generalise versions of diffusion limited aggregation (DLA) in the plane using complex analysis. They showed convergence of the ALE for certain parameters to a single growing slit. Started from a non-trivial initial configuration of k needles and the same parameters, we show that the small-particle scaling limit of ALE is the Laplacian path model, introduced by Carleson and Makarov, in which the tips grow along geodesics towards ∞ . Our proof involves analysis of Loewner’s equation near its singular points, and we extend martingale methods to the backward equation, where what we have to control is non-adapted. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner’s equation at the sharp tips and right-angle bases of slit particles. As an intermediate step we prove a limit result for a model with no such regularisation factor, developing methods which should prove useful in analysing other weakly-regularised models with non-trivial limits.
We consider multifractal Mandelbrot cascades supported on planar C^2 curves with nonvanishing curvature and show that their Fourier dimension is as large as possible, i.e., equal to the infimum of the lower pointwise dimension of the measure.
We provide criteria for Itô integration to behave continuously with respect to Skorokhod’s J_1 and M_1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M_1 setting and unify existing theories in the J_1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M_1 tightness in fact implies J_1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M_1 and J_1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.
In this article, we consider a variety of percolation models on randomly stretched lattices. The first model we study is constructed on the usual square grid ℤ^2 , keeping all vertices untouched while erasing edges according to the following procedure: for every integer i, the entire column of vertical edges contained in the line { x = i } is removed independently of other columns with probability ρ > 0 . Similarly, for every integer j, the entire row of horizontal edges contained in the line { y = j} is removed independently of other rows with probability ρ . On the remaining random lattice, we perform Bernoulli bond percolation. Our main contribution is an alternative proof that the model undergoes a nontrivial phase transition, a result which was earlier established by Hoffman. The main novelty of our work is that the dynamic renormalization employed earlier is now replaced by a static version, which is easier to master and more robust to extend to different models. We emphasize the flexibility of our methods by showing the non-triviality of the phase transition for a new oriented percolation model in a random environment as well as for a model previously investigated by Kesten, Sidoravicius and Vares. In addition, we prove a result about the sensitivity of the phase transition with respect to the stretching mechanism and provide a list of open problems that could be explored using our techniques.
We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a circle of length N with k particles. We show that the mixing time is of order N^2min (k,N-k)^-1/2 , and that the cutoff phenomenon does not occur. This confirms behavior which was separately predicted by Jara, Lacoin and Peres, and it is more broadly believed to hold for integrable models in the KPZ universality class. Our arguments rely on a connection to periodic last passage percolation with a detailed analysis of flat geodesics, as well as a novel random extension and time shift argument for last passage percolation.
We study random walks in random environments generated by the two-dimensional Gaussian free field. More specifically, we consider a rescaled lattice with a small mesh size and view it as a random network where each edge is equipped with an electric resistance given by a regularization for the exponentiation of the Gaussian free field. We prove the tightness of random walks on such random networks at high temperature as the mesh size tends to 0. Our proof is based on a careful analysis of the (random) effective resistances as well as their connections to random walks.
We prove that for an arbitrary indexing group, every ergodic infinitely divisible stationary process that is separable in probability is weakly mixing. This shows that, as in the well-known case of Gaussian stationary processes, ergodicity implies weak mixing is intrinsic to infinite divisibility, removing all structural assumptions on the group from prior results. The main ingredient is a general construction of stochastically continuous extensions for separable in probability stationary processes, reducing the problem to stochastically continuous processes indexed by Polish groups and then to countable groups, where we combine the Maruyama representation with an ergodicity criterion for Poisson suspensions.
We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it is given by the same explicit Gaussian Fixed point obtained in [G. Cannizzaro, Q. Moulard, F. Toninelli, https://arxiv.org/abs/2501.00344, 2025] for the quadratic SBE, but with suitably renormalised coefficients, thereby rigorously justifying and partly correcting the classical Physics derivation of the SBE in [H. van Beijeren, R. Kutner, H. Spohn, Phys. Rev. Lett., 1986] based on Spohn’s theory of nonlinear fluctuating hydrodynamics. Besides, ours is the first universality-type result for out-of-equilibrium systems and the first extension of [M. Hairer, J. Quastel, Forum of Mathematics, Pi, Vol. 6, 2018, e3], to the critical dimension and beyond weak coupling. The major challenge in our work is the mild growth condition on the nonlinearity which renders even the well-posedness of the microscopic equation non-trivial. Additional key novelties include the derivation of fine estimates on the non-quadratic part of the generator as well as a new approximation for the resolvent associated to the solution of the quadratic SBE.
We consider a slight modification of the frog model. For a given graph, each vertex has Poisson(λ ) particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk up to a fixed lifetime t, after which the particle dies and is removed from the system. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters λ and t. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degree and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs.
We consider a parabolic stochastic partial differential equation (SPDE) on [0 ,1] that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an Llog L growth condition. We prove that the SPDE is well posed when the initial data is in L^2[0 ,1]. This solves a strong form of an open problem.
Brox's diffusion is a typical one-dimensional singular diffusion, which was introduced by Brox (1986) as a continuous analogue of Sinai's random walk. In this paper, we will establish quenched heat kernel estimates for short time and annealed heat kernel estimates for large time of Brox's diffusion. The proofs are based on Brox's construction via the scale-transformation and the time-change arguments as well as the theory of resistance forms for symmetric strongly recurrent Markov processes. We emphasize that, since the reference measure of Brox's diffusion does not satisfy the so-called volume doubling conditions neither for the small scale nor the large scale, the existing methods for heat kernel estimates of diffusions in ergodic media do not work, and new techniques will be introduced to establish both quenched and annealed heat kernel estimates of Brox's diffusions, which take into account different oscillation properties for one-dimensional Brownian motion in random environments.
