We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say r, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished r eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy–Widom distributions of the random matrix theory. Especially a phase transition phenomenon is observed. Our results also apply to a last passage percolation model and a queueing model.

The paper considers a stationary point process N in ℝ^d. A point-map picks a point of N in a measurable way. It is called bijective [Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York] if it is generating (by suitable shifts) a bijective mapping on N. Mecke [Math. Nachr. 65 (1975) 335–344] proved that the Palm measure of N is point-stationary in the sense that it is invariant under bijective point-shifts. Our main result identifies this property as being characteristic for Palm measures. This generalizes a fundamental classical result for point processes on the line (see, e.g., Theorem 11.4 in [Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York]) and solves a problem posed in [Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York] and [Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Ann. Inst. H. Poincaré Probab. Statist. 40 141–152]. Our second result guarantees the existence of bijective point-maps that have (almost surely with respect to the Palm measure of N) no fixed points. This answers another question asked by Thorisson. Our final result shows that there is a directed graph with vertex set N that is defined in a translation-invariant way and whose components are almost surely doubly infinite paths. This generalizes and complements one of the main results in [Holroyd, A. E. and Peres, Y. (2003). Electron. Comm. Probab. 8 17–27]. No additional assumptions (as ergodicity, nonlattice type conditions, or a finite intensity) are made in this paper.

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to $$dX_{t}=dW_{t}(X_{t})-\frac{1}{t}\biggl(\int_{0}^{t}\nabla V_{X_{s}}(X_{t})\,ds\biggr)\,dt,$$ where {W_t} is a Brownian vector field on M and V_x(y)=V(x,y) a smooth function. Let $\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\,ds$ denote the normalized occupation measure of X_t. We prove that, when V is symmetric, μ_t converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μ_t converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.

According to a theorem of Schumacher and Brox, for a diffusion X in a Brownian environment, it holds that (X_t−b_logt)/log²t→0 in probability, as t→∞, where b is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for b on an interval [1,x] and study some of the consequences of the computation; in particular, we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a nonrigorous paper by Le Doussal, Monthus and Fisher [Phys. Rev. E 59 (1999) 4795–4840] and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.

This paper elucidates the connection between stationary symmetric α-stable processes with 0<α<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into those corresponding to positive flows and those corresponding to null flows. We show that necessary and sufficient for a stationary stable process to be ergodic is that its positive component vanishes.

Recently, Williams [Bull. London Math. Soc. 34 (2002) 610–612] gave an explicit example of a random time ρ associated with Brownian motion such that ρ is not a stopping time but $\mathbb{E}M_{\rho }=\mathbb{E}M_{0}$ for every bounded martingale M. The aim of this paper is to characterize such random times, which we call pseudo-stopping times, and to construct further examples, using techniques of progressive enlargements of filtrations.

We study an evolution problem in the space of continuous loops in a three-dimensional Euclidean space modeled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids. We establish existence of a local solution starting from Hölder regular loops with index greater than 1/3. When the Hölder regularity of the initial condition X is smaller or equal to 1/2, we require X to be a rough path in the sense of Lyons [Rev. Mat. Iberoamericana 14 (1998) 215–310, System Control and Rough Paths (2002). Oxford Univ. Press]. The solution will then live in an appropriate space of rough paths. In particular, we can construct (local) solution starting from almost every Brownian loop.

For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal {R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k≥1$ $$(\log n)^{k}\Biggl[\frac{1}{n}|\mathcal{R}(n)|+\sum_{j=1}^{k}(-1)^{j}\biggl(\frac{1}{2\pi}\log n+c_{X}\biggr)^{-j}\gamma_{j,n}\Biggr]\to 0\qquad\mbox{a.s.,}$$ where $W_t$ is a Brownian motion, $W^{(n)}_{t}=W_{nt}/\sqrt{n}, γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$ and $c_X$ is a constant depending on the distribution of the random walk.

In a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper we use a new and simplified approach to the lace expansion to prove quite generally that, for finite graphs that are tori, the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph.

The latter is readily verified for several graphs with vertex set {0,1,…,r−1}ⁿ, including the Hamming cube on an alphabet of r letters (the n-cube, for r=2), the n-dimensional torus with nearest-neighbor bonds and n sufficiently large, and the n-dimensional torus with n>6 and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

Suppose X=(X_x,x in Z^d) is a family of i.i.d. variables in some measurable space, B₀ is a bounded set in R^d, and for t>1, H_t is a measure on tB₀ determined by the restriction of X to lattice sites in or adjacent to tB₀. We prove convergence to a white noise process for the random measure on B₀ given by t^−d/2(H_t(tA)−EH_t(tA)) for subsets A of B₀, as t becomes large, subject to H satisfying a “stabilization” condition (whereby the effect of changing X at a single site x is local) but with no assumptions on the rate of decay of correlations. We also give a multivariate central limit theorem for the joint distributions of two or more such measures H_t, and adapt the result to measures based on Poisson and binomial point processes. Applications given include a white noise limit for the measure which counts clusters of critical percolation, a functional central limit theorem for the empirical process of the edge lengths of the minimal spanning tree on random points, and central limit theorems for the on-line nearest-neighbor graph.

In this paper we consider the stochastic recurrence equation Y_t=A_tY_t−1+B_t for an i.i.d. sequence of pairs (A_t,B_t) of nonnegative random variables, where we assume that B_t is regularly varying with index κ>0 and EA_t^κ<1. We show that the stationary solution (Y_t) to this equation has regularly varying finite-dimensional distributions with index κ. This implies that the partial sums S_n=Y₁+⋯+Y_n of this process are regularly varying. In particular, the relation P(S_n>x)∼c₁nP(Y₁>x) as x→∞ holds for some constant c₁>0. For κ>1, we also study the large deviation probabilities P(S_n−ES_n>x), x≥x_n, for some sequence x_n→∞ whose growth depends on the heaviness of the tail of the distribution of Y₁. We show that the relation P(S_n−ES_n>x)∼c₂nP(Y₁>x) holds uniformly for x≥x_n and some constant c₂>0. Then we apply the large deviation results to derive bounds for the ruin probability ψ(u)=P(sup _n≥1((S_n−ES_n)−μn)>u) for any μ>0. We show that ψ(u)∼c₃uP(Y₁>u)μ⁻¹(κ−1)⁻¹ for some constant c₃>0. In contrast to the case of i.i.d. regularly varying Y_t’s, when the above results hold with c₁=c₂=c₃=1, the constants c₁, c₂ and c₃ are different from 1.

An urn model of Diaconis and some generalizations are discussed. A convergence theorem is proved that implies for Diaconis’ model that the empirical distribution of balls in the urn converges with probability one to the uniform distribution.

Given an ergodic dynamical system (X,T,μ), and U⊂X measurable with μ(U)>0, let μ(U)τ_U(x) denote the normalized hitting time of x∈X to U. We prove that given a sequence (U_n) with μ(U_n)→0, the distribution function of the normalized hitting times to U_n converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function F̃, and that in the converging case, $$(⋄)\qquad F(t)=\int_{0}^{t}\bigl(1-\tilde{F}(s)\bigr)\,ds,\qquad t\ge0.$$ This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also.