A noise reinforced Brownian motion is a centered Gaussian process $\hat B=(\hat B(t))_{t\geq 0}$ with covariance $E(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t,$ where $p\in(0,1/2)$ is a reinforcement parameter. Our main purpose is to establish a version of Donsker's invariance principle for a large family of step-reinforced random walks in the diffusive regime, and more specifically, to show that $\hat B$ arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.
We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half line. A reinforced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of $S_N$ (the location of the walker at the $N$^{th}$ step) and $A_N$ the maximum location the walker achieved in $N$ steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its "environment", both parametrized by …
Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex R…
This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian motion. In particular, we proof a conjecture of Mandelbrot that the dimension of the frontier is 4/3. The proof uses a universality principle for conformally invariant measures and a new process, the stochastic Loewner evolution ($SLE$), introduced by Schramm. These ideas can be used to study other planar lattice models from statistical physics at criticality. I d…
National University of Singapore（新加坡国立大学）；The Czech Academy of Sciences, Institute of Information Theory and Automation（捷克科学院信息与自动化研究所）；East China Normal University（华东师范大学）
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models a…
The geometry of coalescing random walks, the Brownian web distance and KPZ universality
Budapest University of Technology and Economics（布达佩斯科技经济大学）；HUN-REN–BME Stochastics Research Group（匈牙利研究网络-布达佩斯科技大学随机过程研究组）；University of Toronto（多伦多大学）
In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq b_n$ at time $t=1$, where $μ_{A_n}=(1/n)\sum_{i}δ_{a_i}, μ_{B_n}=(1/n)\sum_i δ_{b_i}$ are discretization of probability measures $μ_A, μ_B$. Under regularity assumptions of $μ_A, μ_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.
We consider a random planar map $M_n$ which is uniformly distributed over the class of all rooted q-angulations with n faces. We let $\mathbf{m}_n$ be the vertex set of $M_n$, which is equipped with the graph distance $d_\mathrm{gr}$. Both when $q\geq4$ is an even integer and when q=3, there exists a positive constant $c_q$ such that the rescaled metric spaces $(\mathbf{m}_n,c_qn^{-1/4}d_\mathrm{gr})$ converge in distribution in the Gromov-Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.
We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of the empirical measure-valued processes constructed from the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its subsequences as solutions to an integral equation. We use this result to study some universality classes of random matrix flows. These generalize the classical results related to Dyson Br…
Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds. When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be embedded via the Robinson-Schensted-Knuth correspondence a…
This paper is the third chapter of three of the author's undergraduate thesis. In this paper, we study the convergence of local bulk statistics for linearized covariance matrices under Dyson's Brownian motion. We consider deterministic initial data $V$ approximate the Dyson Brownian motion for linearized covariance matrices by the Wigner flow. Using universality results for the Wigner flow, we deduce universality for the linearized covariance matrices. We deduce bulk universality of averaged bulk correlation functions for both biregular bipartite graphs and honest covariance matrices. We also …
For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko-Pastur law and other convergenc…
The spectral heat content is investigated for time-changed killed Brownian motions on C1,1 open sets, where the time change is given by either a subordinator or an inverse subordinator, with the underlying Laplace exponent being regularly varying at \infty with index β\in (0, 1). In the case of inverse subordinators, the asymptotic limit of the spectral heat content is shown to involve a probabilistic term depending only on β\in (0, 1). In contrast, in the case of subordinators, this universality holds only when β\in ( 1/2 , 1).
A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, …
Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions
We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous …
We present a simple model of a random walk with partial memory, which we call the \emph{random memory walk}. We introduce this model motivated by the belief that it mimics the behavior of the once-reinforced random walk in high dimensions and with small reinforcement. We establish the transience of the random memory walk in dimensions three and higher, and show that its scaling limit is a Brownian motion.
Consider non-intersecting Brownian motions on the line leaving from the origin and forced to two arbitrary points. Letting the number of Brownian particles tend to infinity, and upon rescaling, there is a point of bifurcation, where the support of the density of particles goes from one interval to two intervals. In this paper, we show that at that very point of bifurcation a cusp appears, near which the Brownian paths fluctuate like the Pearcey process. This is a universality result within this class of problems. Tracy and Widom obtained such a result in the symmetric case, when the two target…
We consider the one-sided exit problem for (fractionally) integrated random walks and Lévy processes. We prove that the rate of decrease of the non-exit probability -- the so-called survival exponent -- is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractional…
We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from the Wigner semicircle law.
We study a model of $n$ one-dimensional non-intersecting Brownian motions with two prescribed starting points at time $t=0$ and two prescribed ending points at time $t=1$ in a critical regime where the paths fill two tangent ellipses in the time-space plane as $n \to \infty$. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new familiy of limiting determinantal point processes with integrable correlation kernels that are exp…
We show that $\mathbb{P} ( \ell_X(0,T] \leq 1)=(c_X+o(1))T^{-(1-H)}$, where $\ell_X$ is the local time measure at $0$ of any recurrent $H$-self-similar real-valued process $X$ with stationary increments that admits a sufficiently regular local time and $c_X$ is some constant depending only on $X$. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $1-H$ on the decay exponent of $\mathbb{P} ( \ell_X(0,T] \leq 1)$. Our appr…
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log-gas with inverse temperature $β= 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. For $β\not \in \{1, 2, 4\}$, there is no matrix model behind this model, but the statistical physics interpretation of the log-gas is still valid for all $β> 0$. The universality conjecture for invariant ensembles …
We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the a…
We establish an invariance principle corresponding to the universality of random matrices. More precisely, we prove the dynamical universality of random matrices in the sense that, if the random point fields $ \muN $ of $ \nN $-particle systems describing the eigenvalues of random matrices or log-gases with general self-interaction potentials $ \V $ converge to some random point field $ μ$, then the associated natural $ \muN $-reversible diffusions represented by solutions of stochastic differential equations (SDEs) converge to some $ μ$-reversible diffusion given by the solution of an infinit…
In this paper, we prove a universality result of convergence for a bivariate random process defined by the eigenvectors of a sample covariance matrix. Let $V_n=(v_{ij})_{i \leq n,\, j\leq m}$ be a $n\times m$ random matrix, where $(n/m)\to y > 0$ as $ n \to \infty$, and let $X_n=(1/m) V_n V^{*}_n $ be the sample covariance matrix associated to $V_n \:$. Consider the spectral decomposition of $X_n$ given by $ U_n D_n U_n^{*}$, where $U_n=(u_{ij})_{n\times n}$ is an eigenmatrix of $X_n$. We prove, under some moments conditions, that the bivariate random process $<B_{s,t}^{n} = \underset{1\leq j …
We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of the matrices tend to infinity simultaneously. The behavior of the largest log singular values coincides with the large N limit of Dyson Brownian motion with a characteristic drift vector consisting of equally spaced coordinates, which matches the large N limit of the largest log singular values of Brownian motion on GL(N, C). Our method utilizes the formalism of multivariate Bessel generating functions,…
A reinforcement algorithm introduced by H.A. Simon \cite{Simon} produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability $p\in(0,1)$, $\hat U_{n+1}$ is sampled uniformly from $\hat U_1, \ldots, \hat U_n$, and with complementary probability $1-p$, $\hat U_{n+1}$ is a new independent uniform variable. The Glivenko-Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when…
For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension $d\geq 3$, we establish an exact formula for the average number of $(d-1)$-dimensional facets that can be defined by $d$ points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when $d=3$, a case which is of particular interest for applications in biophysics, chemistry and polymer science. We also show that the asymptotical average number of facets behaves as $\langle \mathcal{F}_T^{(d)…
We explain a unified approach to a study of ballistic phase for a large family of self-interacting random walks with a drift and self-interacting polymers with an external stretching force. The approach is based on a recent version of the Ornstein-Zernike theory developed in earlier works. It leads to local limit results for various observables (e.g. displacement of the end-point or number of hits of a fixed finite pattern) on paths of n-step walks (polymers) on all possible deviation scales from CLT to LD. The class of models, which display ballistic phase in the "universality class" discusse…
Consider a time-varying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both c…
We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the…
We consider large-scale point fields which naturally appear in the context of the Kardar-Parisi-Zhang (KPZ) phenomenon. Such point fields are geometrical objects formed by points of mass concentration, and by shocks separating the sources of these points. We introduce similarly defined point fields for processes of coalescing fractional Brownian motions (cfBm). The case of the Hurst index 2/3 is of particular interest for us since, in this case, the power law of the density decay is the same as that in the KPZ phenomenon. In this paper, we present strong numerical evidence that statistical pro…
Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. Simplified models of Wigner's thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential…
We study the ultrametric random matrix ensemble, whose independent entries have variances decaying exponentially in the metric induced by the tree topology on $\mathbb{N}$, and map out the entire localization regime in terms of eigenfunction localization and Poisson statistics. Our results complement existing works on complete delocalization and random matrix universality, thereby proving the existence of a phase transition in this model. In the simpler case of the Rosenzweig-Porter model, the analysis yields a complete characterization of the transition in the local statistics. The proofs are…
In this paper we study mesoscopic fluctuations for Dyson's Brownian motion with $β=2$. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. In this paper we investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that we cons…
We consider branching Brownian motion on the real line with the following selection mechanism: Every time the number of particles exceeds a (large) given number $N$, only the $N$ right-most particles are kept and the others killed. After rescaling time by $\log^3N$, we show that the properly recentred position of the $\lceil αN\rceil$-th particle from the right, $α\in(0,1)$, converges in law to an explicitly given spectrally positive Lévy process. This behaviour has been predicted to hold for a large class of models falling into the universality class of the FKPP equation with weak multiplicat…
This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, imply…
We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $ν$ with a subexponential decay. Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE), re…
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class …
