In this paper we will show how the results found in [Probab. Theory Related Fields 154 (2012) 89–125], about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process or the Hammersley interacting particle process. The method will be to use the well-known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well-known connection between second class particles and competition interfaces.

This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at $d=2$. The cost will be defined in terms of an arbitrary increasing function of the distance.

The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.

An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.

Let $X$ be a locally compact Polish space and let $m$ be a reference Radon measure on $X$. Let $\Gamma_{X}$ denote the configuration space over $X$, that is, the space of all locally finite subsets of $X$. A point process on $X$ is a probability measure on $\Gamma_{X}$. A point process $\mu$ is called determinantal if its correlation functions have the form $k^{(n)}(x_{1},\ldots,x_{n})=\det[K(x_{i},x_{j})]_{i,j=1,\ldots,n}$. The function $K(x,y)$ is called the correlation kernel of the determinantal point process $\mu$. Assume that the space $X$ is split into two parts: $X=X_{1}\sqcup X_{2}$. A kernel $K(x,y)$ is called $J$-Hermitian if it is Hermitian on $X_{1}\times X_{1}$ and $X_{2}\times X_{2}$, and $K(x,y)=-\overline{K(y,x)}$ for $x\in X_{1}$ and $y\in X_{2}$. We derive a necessary and sufficient condition of existence of a determinantal point process with a $J$-Hermitian correlation kernel $K(x,y)$.

We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold.

First, when the driving vector fields satisfy Hörmander’s celebrated “Lie bracket condition,” we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel “deterministic” version of Norris’s lemma for differential equations driven by rough paths. This result, with the added assumption that the linearized equation has moments, will then yield that the transition laws have a smooth density with respect to Lebesgue measure.

Our second main result states that under Hörmander’s condition, the solutions to rough differential equations driven by fractional Brownian motion with $H\in(\frac{1}{3},\frac{1}{2}]$ enjoy a suitable version of the strong Feller property. Under a standard controllability condition, this implies that they admit a unique stationary solution that is physical in the sense that it does not “look into the future.”

Consider a continuous time random walk in $\mathbb{Z} $ with independent and exponentially distributed jumps $\pm1$. The model in this paper consists in an infinite number of such random walks starting from the complement of $\{-m,-m+1,\ldots,m-1,m\}$ at time $-t$, returning to the same starting positions at time $t$, and conditioned not to intersect. This yields a determinantal process, whose gap probabilities are given by the Fredholm determinant of a kernel. Thus this model consists of two groups of random walks, which are contained within two ellipses which, with the choice $m\simeq2t$ to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling $m=\lfloor2t+\sigma t^{1/3}\rfloor$, where parameter $\sigma$ controls the strength of interaction between the two groups of random walkers.

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the $k$th smallest gap, normalized by a factor $n^{-4/3}$, has a limiting density proportional to $x^{3k-1}e^{-x^{3}}$. Concerning the largest gaps, normalized by $n/\sqrt{\log n}$, they converge in ${\mathrm{L} }^{p}$ to a constant for all $p>0$. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion.

Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629–667] suggested “the existence of a form of measure-theoretic duality” between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in “alternating” scenery, thus hinting at a mechanism behind this duality.

Following Cohen and Samorodnitsky [Ann. Appl. Probab. 16 (2006) 1432–1461], we also consider alternating random reward schema associated to random walks at random times. Whereas random reward schema scale to local time fractional stable motions, we show that the alternating random reward schema scale to indicator fractional stable motions.

Finally, we show that one may recursively “subordinate” random time processes to get new local time and indicator fractional stable motions and new stable processes in random scenery or at random times. When $\alpha=2$, the fractional stable motions given by the recursion are fractional Brownian motions with dyadic $H\in(0,1)$. Also, we see that “un-subordinating” via a time-change allows one to, in some sense, extract Brownian motion from fractional Brownian motions with $H<1/2$.

We prove that an adequately rescaled sequence $\{F_{n}\}$ of self-adjoint operators, living inside a fixed free Wigner chaos of even order, converges in distribution to a centered free Poisson random variable with rate $\lambda>0$ if and only if $\varphi(F_{n}^{4})-2\varphi(F_{n}^{3})\rightarrow2\lambda^{2}-\lambda$ (where $\varphi$ is the relevant tracial state). This extends to a free setting some recent limit theorems by Nourdin and Peccati [Ann. Probab. 37 (2009) 1412–1426] and provides a noncentral counterpart to a result by Kemp et al. [Ann. Probab. 40 (2012) 1577–1635]. As a by-product of our findings, we show that Wigner chaoses of order strictly greater than 2 do not contain nonzero free Poisson random variables. Our techniques involve the so-called “Riordan numbers,” counting noncrossing partitions without singletons.

In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. These functionals appear as coefficients in the probability content of a tube around a convex set $D\subset\mathbb{R}^{k}$ under the standard Gaussian law $N(0,I_{k\times k})$. Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be expressed in terms of reasonably smooth Wiener functionals.

In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten’s condition [Acta Math. 131 (1973) 207–248] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise sequence of the equations can have light tails. The results of the paper are analogs to those obtained by A. V. Nagaev [Theory Probab. Appl. 14 (1969) 51–64; 193–208] and S. V. Nagaev [Ann. Probab. 7 (1979) 745–789] in the case of partial sums of i.i.d. random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies in the sequence. We apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model.

In this paper, we show that reflecting Brownian motion in any bounded domain $D$ can be approximated, as $k\to\infty$, by simple random walks on “maximal connected” subsets of $(2^{-k}\mathbb{Z} ^{d})\cap D$ whose filled-in interiors are inside of $D$.

The purpose of this paper is to establish the convergence in law of the sequence of “midpoint” Riemann sums for a stochastic process of the form $f'(W)$, where $W$ is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-of-variable formula in law with a second order correction term which is an Itô integral of $f"(W)$ with respect to a Gaussian martingale independent of $W$. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for $q$-fold Skorohod integrals, which is a multi-dimensional extension of a result proved by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39–64]. The results proved in this paper are generalizations of previous work by Swanson [Ann. Probab. 35 (2007) 2122–2159] and Nourdin and Réveillac [Ann. Probab. 37 (2009) 2200–2230], who found a similar formula for two particular types of bifractional Brownian motion. We provide three examples of Gaussian processes $W$ that meet the necessary covariance bounds. The first one is the bifractional Brownian motion with parameters $H\le1/2$, $HK=1/4$. The others are Gaussian processes recently studied by Swanson [Probab. Theory Related Fields 138 (2007) 269–304], [Ann. Probab. 35 (2007) 2122–2159] in connection with the fluctuation of empirical quantiles of independent Brownian motion. In the first example the Gaussian martingale is a Brownian motion, and expressions are given for the other examples.

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 $(\mathbf{m}_{n},c_{q}n^{-1/4}d_{\mathrm{gr}})$ 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 the Burgers equation on the real line with forcing given by Poissonian noise with no periodicity assumption. Under a weak concentration condition on the driving random force, we prove existence and uniqueness of a global solution in a certain class. We describe its basin of attraction that can also be viewed as the main ergodic component for the model. We establish existence and uniqueness of global minimizers associated to the variational principle underlying the dynamics. We also prove the diffusive behavior of the global minimizers on the universal cover in the periodic forcing case.

The star–triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.

We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter $H>1/4$. We remark on the relevance of such estimates to a number of significant open problems.