We consider the Poisson Boolean model of continuum percolation. We show that there is a subcritical phase if and only if E(R^d) is finite, where R denotes the radius of the balls around Poisson points and d denotes the dimension. We also give related results concerning the integrability of the diameter of subcritical clusters.

Stochastic Loewner evolutions (SLE) with a multiple $\sqrt{\kappa}B$ of Brownian motion B as driving process are random planar curves (if κ≤4) or growing compact sets generated by a curve (if κ>4). We consider here more general Lévy processes as driving processes and obtain evolutions expected to look like random trees or compact sets generated by trees, respectively. We show that when the driving force is of the form $\sqrt{\kappa}B+\theta^{1/\alpha}S$ for a symmetric α-stable Lévy process S, the cluster has zero or positive Lebesgue measure according to whether κ≤4 or κ>4. We also give mathematical evidence that a further phase transition at α=1 is attributable to the recurrence/transience dichotomy of the driving Lévy process. We introduce a new class of evolutions that we call α-SLE. They have α-self-similarity properties for α-stable Lévy driving processes. We show the phase transition at a critical coefficient θ=θ₀(α) analogous to the κ=4 phase transition.

Let $\mathcal{C}_{1}$ denote the largest connected component of the critical Erdős–Rényi random graph $G(n,{\frac{1}{n}})$. We show that, typically, the diameter of $\mathcal{C}_{1}$ is of order n^1/3 and the mixing time of the lazy simple random walk on $\mathcal{C}_{1}$ is of order n. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size n^2/3 of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d−1)≤1+O(n^−1/3).

We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, for example, displacement, over such chains obey classical limit laws, and our construction leads to an effective random walk representation of percolation clusters.

The results include a derivation of a sharp Ornstein–Zernike type asymptotic formula for two point functions, a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling.

In two dimensions duality considerations enable a reformulation of these results for supercritical nearest-neighbor random cluster measures, in particular, for nearest-neighbor Potts models in the phase transition regime. Accordingly, we prove that in two dimensions Potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive. Thus, no roughening transition is possible in the whole regime where our results apply.

Our results hold under an assumption of exponential decay of finite volume wired connectivities [assumption (1.2) below] in rectangular domains that is conjectured to hold in the whole subcritical regime; the latter is known to be true, in any dimensions, when q=1, q=2, and when q is sufficiently large. In two dimensions assumption (1.2) holds whenever there is an exponential decay of connectivities in the infinite volume measure. By duality, this includes all supercritical nearest-neighbor Potts models with positive surface tension between ordered phases.

We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the trap model in the complete graph, and subsequently derive aging results for those models in this context.

Consider a uniformly random regular graph of a fixed degree d≥3, with n vertices. Suppose that each edge is open (closed), with probability p(q=1−p), respectively. In 2004 Alon, Benjamini and Stacey proved that p^*=(d−1)⁻¹ is the threshold probability for emergence of a giant component in the subgraph formed by the open edges. In this paper we show that the transition window around p^* has width roughly of order n^−1/3. More precisely, suppose that p=p(n) is such that ω:=n^1/3|p−p^*|→∞. If p<p^*, then with high probability (whp) the largest component has O((p−p^*)⁻²log n) vertices. If p>p^*, and log ω≫log log n, then whp the largest component has about n(1−(pπ+q)^d)≍n(p−p^*) vertices, and the second largest component is of size (p−p^*)⁻²(log n)^1+o(1), at most, where π=(pπ+q)^d−1, π∈(0, 1). If ω is merely polylogarithmic in n, then whp the largest component contains n^2/3+o(1) vertices.

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.

Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ≥0. We prove that, with probability one, the Hausdorff dimension of γ is equal to Min(2, 1+κ/8).

We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure μ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with μ.

We prove that the chordal SLE_κ trace is reversible for κ∈(0, 4].

We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.

We investigate the free energy of nearest-neighbor random walks on ℤ^d, endowed with a drift along the first axis and evolving in a nonnegative random potential given by i.i.d. random variables. Our main result concerns the ballistic regime in dimensions d≥4, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.

We introduce a new version of Stein’s method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of “local” itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by P. Bickel.