Showing papers in "Probability Theory and Related Fields in 2013"

The extremal process of branching Brownian motion

Abstract: We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses.

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Abstract: Slepian and Sudakov–Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Levy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

Branching Brownian motion seen from its tip

Abstract: It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, A branching random walk seen from the tip, 2010, Poissonian statistics in the extremal process of branching Brownian motion, 2010; Arguin et al., The extremal process of branching Brownian motion, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (The extremal process of branching Brownian motion, 2011).

A contour line of the continuum Gaussian free field

Abstract: Consider an instance $$h$$ of the Gaussian free field on a simply connected planar domain $$D$$ with boundary conditions $$-\lambda $$ on one boundary arc and $$\lambda $$ on the complementary arc, where $$\lambda $$ is the special constant $$\sqrt{\pi /8}$$ . We argue that even though $$h$$ is defined only as a random distribution, and not as a function, it has a well-defined zero level line $$\gamma $$ connecting the endpoints of these arcs, and the law of $$\gamma $$ is $$\mathrm{SLE}(4)$$ . We construct $$\gamma $$ in two ways: as the limit of the chordal zero contour lines of the projections of $$h$$ onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of $$h, \gamma $$ is “local” (it does not change when $$h$$ is modified away from $$\gamma $$ ) and derive some general properties of local sets.

Eigenvector Distribution of Wigner Matrices

Abstract: We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure ν ij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution ν ij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

Limits of spiked random matrices I

Abstract: Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik et al. (Ann Probab 33:1643–1697, 2005). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrodinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2,4, yielding in particular a new and simple proof of the Painleve representations for these Tracy–Widom distributions.

Outliers in the spectrum of iid matrices with bounded rank perturbations

Abstract: It is known that if one perturbs a large iid random matrix by a bounded rank error, then the majority of the eigenvalues will remain distributed according to the circular law. However, the bounded rank perturbation may also create one or more outlier eigenvalues. We show that if the perturbation is small, then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation; but if the perturbation is large, then many more outliers can be created, and their law is governed by the zeroes of a random Laurent series with Gaussian coefficients. On the other hand, these outliers may be eliminated by enforcing a row sum condition on the final matrix.

Invariance principle for the random conductance model

Abstract: We study a continuous time random walk X in an environment of iid random conductances $${\mu_{e} \in [0,\infty)}$$ in $${\mathbb{Z}^d}$$ We assume that $${\mathbb{P}(\mu_{e} > 0) > p_c}$$ , so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μ e We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality

Optimal rates of convergence for estimating Toeplitz covariance matrices

Abstract: Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In this paper, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm. The properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound. The results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces. The minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent Gaussian scale model and through a careful selection of a finite collection of least favorable parameters. In addition, optimal rate of convergence for estimating the inverse of a Toeplitz covariance matrix is also established.

The Bernstein–Orlicz norm and deviation inequalities

Abstract: We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.

Rough Burgers-like equations with multiplicative noise

Abstract: We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space–time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.

Convergence rates for rank-based models with applications to portfolio theory

Abstract: We determine rates of convergence of rank-based interacting diffusions and semimartingale reflecting Brownian motions to equilibrium. Bounds on fluctuations of additive functionals are obtained using Transportation Cost-Information inequalities for Markov processes. We work out various applications to the rank-based abstract equity markets used in Stochastic Portfolio Theory. For example, we produce quantitative bounds, including constants, for fluctuations of market weights and occupation times of various ranks for individual coordinates. Another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio. This produces estimates of probabilities of “beating the market”.

Fixed points of the smoothing transform: two-sided solutions

Abstract: Given a sequence (C, T) = (C, T1, T2, . . .) of real-valued random variables with Tj ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : Tj g 0}, the smoothing transform associated wi ...

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Abstract: Consider an $$n \times n$$ Hermitian random matrix with, above the diagonal, independent entries with $$\alpha $$ -stable symmetric distribution and $$0 < \alpha < 2$$ . We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as $$n$$ goes to infinity. When $$1 < \alpha < 2$$ and $$ p > 2$$ , we give vanishing bounds on the $$L^p$$ -norm of the eigenvectors normalized to have unit $$L^2$$ -norm. On the contrary, when $$0 < \alpha < 2/3$$ , we prove that these eigenvectors are localized.

On the chaotic character of the stochastic heat equation, II

Abstract: Consider the stochastic heat equation $${\partial_t u = (\varkappa/2)\Delta u+\sigma(u)\dot{F}}$$ , where the solution u := u t (x) is indexed by $${(t, x) \in (0, \infty) \times {\bf R}^d}$$ , and $${\dot{F}}$$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- $${\|x\|}$$ fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is $${f(x)\propto \|x\|^{-\alpha}}$$ , then the “fluctuation exponents” of the solution are $${\psi}$$ for the spatial variable and $${2\psi-1}$$ for the time variable, where $${\psi:=2/(4-\alpha)}$$ . Moreover, these exponent relations hold as long as $${\alpha \in (0, d \wedge 2)}$$ ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987].

Tree-valued resampling dynamics Martingale problems and applications

Abstract: The measure-valued Fleming–Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the “individuals” in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming–Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled “individuals”. We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming–Viot dynamics.

Precise large deviations for dependent regularly varying sequences

Abstract: We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.

Gumbel fluctuations for cover times in the discrete torus

Abstract: This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in Aldous and Fill (Reversible Markov chains and random walks on graphs, in preparation). We also derive some corollaries which qualitatively describe “how” covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman (Ann Math (2) 171(3):2039–2087, 2010), and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.

Asymptotic equivalence for nonparametric regression with non-regular errors

Abstract: Asymptotic equivalence in Le Cam’s sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size n and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.

Matchings on infinite graphs

Abstract: Elek and Lippner (Proc. Am. Math. Soc. 138(8), 2939-2947, 2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erdős-Renyi random graphs.

Mobile geometric graphs: detection, coverage and percolation

Abstract: We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in $${\mathbb{R}^d}$$ with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.

Planar diffusions with rank-based characteristics and perturbed Tanaka equations

Abstract: For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffusion process (X 1(·), X 2(·)) with values in the plane and infinitesimal generator $${\begin{array}{ll}\fancyscript{L}=\mathbf{1}_{\{ x_1 > x_2\}}\left(\frac{\rho^2}2{\frac{\partial^2}{\partial x{_1^2}}} +\frac{\sigma^2}{2}{\frac{\partial^2}{\partial x{_2^2}}}-h\frac{\partial}{\partial x_1} +g \frac{\partial}{\partial{x_2}}\right)\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathbf{1}_{\{ x_1\le x_2\}}\left(\frac{\sigma^2}{2}{\frac{\partial^2}{\partial x{_1^2}}} +\frac{\rho^2}{2} {\frac{\partial^2}{\partial x{_2^2}}}+g\frac{\partial}{\partial x_1} - h \frac{\partial}{\partial{x_2}}\right),\,\,\,\,\,\,\,\,\,\,\,\, (0.1)\end{array}}$$ and discuss its realization in terms of appropriate systems of stochastic differential equations. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation $$\begin{array}{ll}{\rm d}Z(t) = f \big(Z (t)\big){\rm d}M( t) + {\rm d}N(t), \quad Z(0) = \xi\end{array}$$ driven by suitable continuous, orthogonal semimartingales M(·) and N(·) and with f(·) of bounded variation, which we study here in detail; and those of a one-dimensional diffusion Y(·) with bang-bang drift $${dY(t) = -\lambda {\rm sign} \big( Y (t) \big) {\rm d}t + {\rm d}W (t), Y(0)=y}$$ driven by a standard Brownian motion W(·). We also show that the planar diffusion (X 1(·), X 2(·)) can be represented in terms of this process Y(·), its local time L Y (·) at the origin, and an independent standard Brownian motion Q(·), in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.

Hadamard's formula and couplings of SLEs with free field

Abstract: The relation between level lines of Gaussian free fields (GFF) and SLE4-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE4 is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.

Improved mixing condition on the grid for counting and sampling independent sets

Abstract: The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set I in a graph G is weighted proportionally to λ|I|, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree Δ ≥ 3, is a well known computationally challenging problem. More concretely, let \({\lambda_c(\mathbb{T}_\Delta)}\) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite Δ-regular tree; recent breakthrough results of Weitz (Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 140–149, 2006) and Sly (Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 287–296, 2010) have identified \({\lambda_c(\mathbb{T}_\Delta)}\) as a threshold where the hardness of estimating the above partition function undergoes a computational transition. We focus on the well-studied particular case of the square lattice \({\mathbb{Z}^2}\) , and provide a new lower bound for the uniqueness threshold, in particular taking it well above \({\lambda_c(\mathbb{T}_4)}\) . Our technique refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hard-core model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to \({\mathbb{Z}^2}\) we prove that strong spatial mixing holds for all λ < 2.3882, improving upon the work of Weitz that held for λ < 27/16 = 1.6875. Our results imply a fully-polynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core distribution.

Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field

Abstract: We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an \(o(\log n)\) bound by Chatterjee (Chaos, concentration, and multiple valleys, 2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (Commun. Pure Appl. Math, 2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant.

Differentiability at the edge of the percolation cone and related results in first-passage percolation

Abstract: We study first-passage percolation in two dimensions, using measures μ on passage times with b: = inf supp(μ) > 0 and \({\mu(\{b\})=p\geq \vec p_c}\) , the threshold for oriented percolation. We first show that for each such μ, the boundary of the limit shape for μ is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if μ is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman–Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for μ. This result confirms a prediction of Newman and Piza (Ann Probab 23:977–1005, 1995) and Zhang (Ann Probab 36:331–362, 2008). Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents χ and ξ.

“Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth

Abstract: We obtain a representation of Feller’s branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H at its current level. As in the classical Ray–Knight representation, the excursions of H are the exploration paths of the trees of descendants of the ancestors at time t = 0, and the local time of H at height t measures the population size at time t. We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time s and living at time t = H s is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating H with a sequence of Harris paths H N which figure in a Ray–Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of H N together with its local times and with the Girsanov densities that introduce the dependence in the reproduction.

Strong solutions of stochastic equations with rank-based coefficients

Abstract: We study finite and countably infinite systems of stochastic differential equations, in which the drift and diffusion coefficients of each component (particle) are determined by its rank in the vector of all components of the solution. We show that strong existence and uniqueness hold until the first time three particles collide. Motivated by this result, we improve significantly the existing conditions for the absence of such triple collisions in the case of finite-dimensional systems, and provide the first condition of this type for systems with a countable infinity of particles.

A simple method for finite range decomposition of quadratic forms and Gaussian fields

Abstract: We present a simple method to decompose the Green forms corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green form. This result gives rise to multiscale decompositions of the associated Gaussian free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Our method makes use of the finite propagation speed of the wave equation and Chebyshev polynomials. It improves several existing results and also gives simpler proofs.

Concentration and convergence rates for spectral measures of random matrices

Abstract: The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.