Turn the set of permutations of n objects into a graph G_n by connecting two permutations that differ by one transposition, and let σ_t be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17–26] showed that the limiting behavior of the distance from the identity at time cn/2 has a phase transition at c=1. Here we investigate some consequences of this result for the geometry of G_n. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occurs at a critical radius equal to n/4. Let T be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius an for a constant 0<a<1. Then when a<1/4, if the geodesics are suitably chosen, with high probability T is δ-thin for some δ>0, whereas it is always O(n)-thick when a>1/4. We also show that the hitting distribution of the sphere of radius an is asymptotically singular with respect to the uniform distribution. Finally, we prove that the critical behavior of this Gromov-like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius an. However, in this case, the critical radius is a=1−log2.

A random composition of n appears when the points of a random closed set ℛ̃⊂[0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts K_n of this composition and other related functionals under the assumption that ℛ̃=ϕ(S_•), where (S_t,t≥0) is a subordinator and ϕ:[0,∞]→[0,1] is a diffeomorphism. We derive the asymptotics of K_n when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function ϕ(x)=1−e^−x, we establish a connection between the asymptotics of K_n and the exponential functional of the subordinator.

There are n queues, each with a single server. Customers arrive in a Poisson process at rate λn, where 0<λ<1. Upon arrival each customer selects d≥2 servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1. We show that the system is rapidly mixing, and then investigate the maximum length of a queue in the equilibrium distribution. We prove that with probability tending to 1 as n→∞ the maximum queue length takes at most two values, which are lnlnn/lnd+O(1).

We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ℝ^d. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension d tends to infinity.

In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward ɛ-movability which will be a key tool for our analysis.

Let 0<a<b<∞ be fixed scalars. Assign independently to each edge in the lattice ℤ² the value a with probability p or the value b with probability 1−p. For all u,v∈ℤ², let T(u,v) denote the first passage time between u and v. We show that there are points x∈ℝ² such that the “time constant” in the direction of x, namely, lim _n→∞n⁻¹E_p[T(0,nx)], is not a three times differentiable function of p.

Assign to each site of the integer lattice ℤ^d a real score, sampled according to the same distribution F, independently of the choices made at all other sites. A lattice animal is a finite connected set of sites, with its weight being the sum of the scores at its sites. Let N_n be the maximal weight of those lattice animals of size n that contain the origin. Denote by N the almost sure finite constant limit of n⁻¹N_n, which exists under a mild condition on the positive tail of F. We study certain geometrical aspects of the lattice animal with maximal weight among those contained in an n-box where n is large, both in the supercritical phase where N>0, and in the critical case where N=0.

A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.

We develop a new method to uniquely solve a large class of heat equations, so-called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock–Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier–Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.

The background for the general mathematical link between utility and information theory investigated in this paper is a simple financial market model with two kinds of small traders: less informed traders and insiders, whose extra information is represented by an enlargement of the other agents’ filtration. The expected logarithmic utility increment, that is, the difference of the insider’s and the less informed trader’s expected logarithmic utility is described in terms of the information drift, that is, the drift one has to eliminate in order to perceive the price dynamics as a martingale from the insider’s perspective. On the one hand, we describe the information drift in a very general setting by natural quantities expressing the probabilistic better informed view of the world. This, on the other hand, allows us to identify the additional utility by entropy related quantities known from information theory. In particular, in a complete market in which the insider has some fixed additional information during the entire trading interval, its utility increment can be represented by the Shannon information of his extra knowledge. For general markets, and in some particular examples, we provide estimates of maximal utility by information inequalities.

We study a class of one-dimensional interacting particle systems with random boundaries as a microscopic model for Stefan’s melting and freezing problem. We prove that under diffusive rescaling these particle systems exhibit a hydrodynamic behavior described by the solution of a Cauchy–Stefan problem.

By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11—with the former number applying when the metric used to define mixing is the total variation distance and the latter when it is the separation distance. This and other related results assume all 52 cards in the deck to be distinct and require all 52! permutations of the deck to be almost equally likely for the deck to be considered well mixed. In many instances, not all cards in the deck are distinct and only the sets of cards dealt out to players, and not the order in which they are dealt out to each player, needs to be random. We derive transition probabilities under riffle shuffles between decks with repeated cards to cover some instances of the type just described. We focus on decks with cards all of which are labeled either 1 or 2 and describe the consequences of having a symmetric starting deck of the form 1,…,1,2,…,2 or 1,2,…,1,2. Finally, we consider mixing times for common card games.