We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence (O(n)) of outlet variables, the nth of which gives the number of outlets in the box centered at the origin of side length 2ⁿ. The most important of these properties describes the sequence’s renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers for (O(n)). We then show consequences of these limit theorems for the pond radii and outlet weights.

We study the droplet that results from conditioning the planar subcritical Fortuin–Kasteleyn random cluster model on the presence of an open circuit Γ₀ encircling the origin and enclosing an area of at least (or exactly) n². We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Γ₀), this being the maximum distance from a point in the circuit Γ₀ to the boundary ∂ conv(Γ₀) of the circuit’s convex hull; and in a longitudinal sense by what we term maximum facet length, namely, the length of the longest line segment of which the polygon ∂ conv(Γ₀) is formed. We prove that there exists a constant c > 0 such that the conditional probability that the normalised quantity n^−1/3(log n)^−2/3 MLR(Γ₀) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n^−2/3(log n)^−1/3 MFL(Γ₀). To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in [14], the fluctuations MLR(Γ₀) and MFL(Γ₀) are determined up to a constant factor.

Let $\mathcal{C}_{1}$ be the largest component of the Erdős–Rényi random graph $\mathcal{G}(n,p)$. The mixing time of random walk on $\mathcal{C}_{1}$ in the strictly supercritical regime, p = c/n with fixed c > 1, was shown to have order log²n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p = (1 + ε)/n where λ = ε³n is bounded, Nachmias and Peres proved that the mixing time on $\mathcal{C}_{1}$ is of order n. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of $\mathcal{C}_{1}$ in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim.

In this paper, we show that for p = (1 + ε)/n with λ = ε³n → ∞ and λ = o(n), the mixing time on $\mathcal{C}_{1}$ is with high probability of order (n/λ)log²λ. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., p = (1 − ε)/n with λ as above].

We prove a Girsanov identity on the Poisson space for anticipating transformations that satisfy a strong quasi-nilpotence condition. Applications are given to the Girsanov theorem and to the invariance of Poisson measures under random transformations. The proofs use combinatorial identities for the central moments of Poisson stochastic integrals.

In this paper, a Feynman–Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman–Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman–Kac integral is the weak solution to the stochastic partial differential equation.

This paper develops asymptotic approximations of P(∫_Te^f(t) dt > b) as b → ∞ for a homogeneous smooth Gaussian random field, f, living on a compact d-dimensional Jordan measurable set T. The integral of an exponent of a Gaussian random field is an important random variable for many generic models in spatial point processes, portfolio risk analysis, asset pricing and so forth.

The analysis technique consists of two steps: 1. evaluate the tail probability P(∫_Ξe^f(t) dt > b) over a small domain Ξ depending on b, where mes(Ξ) → 0 as b → ∞ and mes(⋅) is the Lebesgue measure; 2. with Ξ appropriately chosen, we show that P(∫_Te^f(t) dt > b) = (1 + o(1)) mes(T) mes⁻¹(Ξ) P(∫_Ξe^f(t) dt > b).

In a recent paper [Statist. Probab. Lett. 78 (2008) 1711–1721] it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function ₂F₁. In the present paper we fully exploit a formula due to Thomae [J. Reine Angew. Math. 87 (1879) 26–73] in order to generalize substantially the class of random continuous fractions with a density of the above form. This involves the design of seven particular graphs. Infinite paths on them lead to random continued fractions with an explicit distribution. A careful study about the set of five real parameters leading to a beta-hypergeometric distribution is required, relying on almost forgotten results mainly due to Felix Klein.

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn–Kalai–Linial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small “influence sum” are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in ℝⁿ of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on ℝⁿ and the class of sets invariant under transitive permutation group of the coordinates.

We present a construction of a Lévy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov theorem. We also extend the pruning procedure to this super-critical case. Let ψ be a critical branching mechanism. We set ψ_θ(⋅) = ψ(⋅ + θ) − ψ(θ). Let Θ = (θ_∞, +∞) or Θ = [θ_∞, +∞) be the set of values of θ for which ψ_θ is a conservative branching mechanism. The pruning procedure allows to construct a decreasing Lévy-CRT-valued Markov process $({\mathcal{T}}_{\theta},\theta\in\Theta)$, such that ${\mathcal{T}}_{\theta}$ has branching mechanism ψ_θ. It is sub-critical if θ > 0 and super-critical if θ < 0. We then consider the explosion time A of the CRT: the smallest (negative) time θ for which the continuous state branching process (CB) associated with ${\mathcal{T}}_{\theta}$ has finite total mass (i.e., the length of the excursion of the exploration process that codes the CRT is finite). We describe the law of A as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to A. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton–Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CB behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous’s CRT.

We prove equivalent conditions for two-sided sub-Gaussian estimates of heat kernels on metric measure spaces.

We study the eigenvalues of the covariance matrix 1/n M^∗M of a large rectangular matrix M = M_n,p = (ζ_ij)_{1≤i≤p;1≤j≤n} whose entries are i.i.d. random variables of mean zero, variance one, and having finite C₀th moment for some sufficiently large constant C₀.

The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549–572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions.

As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C₀th moment rather than exponential decay.

In this paper we study the homogenization of a nonautonomous parabolic equation with a large random rapidly oscillating potential in the case of one-dimensional spatial variable. We show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables, then, under proper mixing assumptions, the limit equation is deterministic, and convergence in probability holds. To the contrary, for the potential having a microstructure only in one of these variables, the limit problem is stochastic, and we only have convergence in law.

We give a second look at stationary stable processes by interpreting the self-similar property at the level of the Lévy measure as characteristic of a Maharam system. This allows us to derive structural results and their ergodic consequences.