We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the structure of the tiling at the center of the Aztec diamond.

Let Π be an ergodic simple point process on ℝ^d and let Π^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057–2064] proved that there exists a shift coupling of Π and Π^*; that is, one can select a (random) point Y of Π such that translating Π by −Y yields a configuration whose law is that of Π^*. We construct shift couplings in which Y and Π^* are functions of Π, and prove that there is no shift coupling in which Π is a function of Π^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of ℝ^d) to the points of Π. The construction is based on the Gale–Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9–15]. Next, let Γ be an ergodic random element of {0,1}^ℤd and let Γ^* be Γ conditioned on Γ(0)=1. A shift coupling X of Γ and Γ^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of Γ if and only if the marginal E[Γ(0)] is the reciprocal of an integer. When the law of Γ is product measure and d≥3, we prove that there exists an extra head scheme X satisfying Eexpc‖X‖^d<∞; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405–1425].

We consider one-dimensional systems of self-gravitating sticky particles with random initial data and describe the process of aggregation in terms of the largest cluster size L_n at any fixed time prior to the critical time. The asymptotic behavior of L_n is also analyzed for sequences of times tending to the critical time. A phenomenon of phase transition shows up, namely, for small initial particle speeds (“cold” gas) L_n has logarithmic order of growth while higher speeds (“warm” gas) yield polynomial rates for L_n.

We study systems of particles on a line which have a maximum, are locally finite and evolve with independent increments. “Quasi-stationary states” are defined as probability measures, on the σ-algebra generated by the gap variables, for which joint distribution of gaps between particles is invariant under the time evolution. Examples are provided by Poisson processes with densities of the form ρ(dx)=e^−sxs dx, with s>0, and linear superpositions of such measures. We show that, conversely, any quasi-stationary state for the independent dynamics, with an exponentially bounded integrated density of particles, corresponds to a superposition of Poisson processes with densities ρ(dx)=e^−sxs dx with s>0, restricted to the relevant σ-algebra. Among the systems for which this question is of some relevance are spin-glass models of statistical mechanics, where the point process represents the collection of the free energies of distinct “pure states,” the time evolution corresponds to the addition of a spin variable and the Poisson measures described above correspond to the so-called REM states.

Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom, we calculate and give an interpretation of linearization coefficients for a number of polynomial families. The processes involved may have independent, freely independent or q-independent increments. The use of noncommutative stochastic processes extends the range of applications significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier and Rogers and continuous big q-Hermite polynomials.

We also show that the q-Poisson process is a Markov process.

This paper concerns the Cauchy problem in R^d for the stochastic Navier–Stokes equation $$∂_t\mathbf u=Δ\mathbf u−(\mathbf u,∇)\mathbf u−∇p+\mathbf f(\mathbf u)+[(σ,∇)\mathbf u−∇p̃+\mathbf g(\mathbf u)]○\dot W,\mathbf u(0)=\mathbf u_0, \mathrm{div} \mathbf u=0,$$ driven by white noise Ẇ. Under minimal assumptions on regularity of the coefficients and random forces, the existence of a global weak (martingale) solution of the stochastic Navier–Stokes equation is proved. In the two-dimensional case, the existence and pathwise uniqueness of a global strong solution is shown. A Wiener chaos-based criterion for the existence and uniqueness of a strong global solution of the Navier–Stokes equations is established.

We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting behavior of quadratic functionals of Gaussian processes.

This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+β)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu=u^1+β.

Various features of the two-parameter family of Schramm–Loewner evolutions SLE (κ,ρ) are studied. In particular, we derive certain restriction properties that lead to a “strong duality” conjecture, which is an identity in law between the outer boundary of a variant of the SLE (κ) process for κ≥4 and a variant of the SLE (16/κ) process.

We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation $$dX_{t}=-\nabla F(X_{t})\,dt+\sqrt{2\varepsilon }\,dW_{t},\qquad \varepsilon >0,$$ and the spectrum near zero of its generator −L_ɛ≡ɛΔ−∇F⋅∇, where F:ℝ^d→ℝ and W denotes Brownian motion on ℝ^d. For generic F to each local minimum of F there corresponds a metastable state. We prove that the distribution of its rescaled relaxation time converges to the exponential distribution as ɛ↓0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_ɛ with eigenvalue which converges to zero exponentially fast in 1/ɛ. Modulo errors of exponentially small order in 1/ɛ this eigenvalue is given as the inverse of the expected metastable relaxation time. The eigenstate is highly concentrated in the basin of attraction of the corresponding trap.

Let X_i, i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a smooth enough mapping from B into R. An asymptotic evaluation of Z_n=E(exp(nΦ(∑_i=1ⁿX_i/n))), up to a factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields 72 (1986) 305–318] and Kusuoka and Liang [Probab. Theory Related Fields 116 (2000) 221–238]. In this paper, a detailed asymptotic expansion of Z_n as n→∞ is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.

We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint distribution of the number of connected components, of the sizes of the giant components and of the numbers of the excess edges of the giant components. For the supercritical case, we obtain the asymptotics of normal deviations and the logarithmic asymptotics of large and moderate deviations of the joint distribution of the number of components, of the size of the largest component and of the number of the excess edges of the largest component. For the critical case, we obtain the logarithmic asymptotics of moderate deviations of the joint distribution of the sizes of connected components and of the numbers of the excess edges. Some related asymptotics are also established. The proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory. As a byproduct of the results, we provide some additional insight into the nature of phase transitions in sparse random graphs.

This paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on X₁,…,X_Nn where X₁,X₂,… are i.i.d. in ℜ² with distribution F and N_n is independent of the X_i and satisfies N_n/n→_p1. A new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which F has unbounded support. The general results are applied to a number of special cases, including elliptically contoured distributions, distributions with independent Weibull-like margins and distributions with parallel level curves.