Wetting of gradient fields: pathwise estimates

We consider the wetting transition in the framework of an effective interface model of gradient type, in dimension 2 and higher. We prove pathwise estimates showing that the interface is localized in the whole thermodynamically-defined partial wetting regime considered in earlier works. Moreover, we study how the interface delocalizes as the wetting transition is approached. Our main tool is reflection positivity in the form of the chessboard estimate.Comment: Some typos removed after proofreading. Version to be published in PTR

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in ℝ d is supported on a set of Hausdorff dimension strictly less thand [2]. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of ℤ d ,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)N(d,β), anyx ∈ ℤ d , and anyA ⊂ {1, ,n} d •{y∈ℤ whereν A,x (y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ →

Estimates for Dirichlet Eigenfunctions

Estimates for the Dirichlet eigenfunctions near the boundary of an open, bounded set in euclidean space are obtained. It is assumed that the boundary satisfies a uniform capacitary density conditio

On the speed of convergence in Strassen's law of the iterated logarithm

Here there is derived a condition on sequences $\varepsilon_n \downarrow 0$ which implies that $P\lbrack W(n^\bullet)/(2n \log \log n)^\frac{1}{2} \not\in K^\varepsilon n \mathrm{i.o.}\rbrack = 0$, where $W$ is the Wiener process and $K$ is the compact set in Strassen's law of the iterated logarithm. A similar result for random walks is also given

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

Heat content and inradius for regions with a Brownian boundary

In this paper we consider \beta [0, s], Brownian motion of time length s > 0, in m-dimensional Euclidean space R^m and on the m-dimensional torus T^m. We compute the expectation of (i) the heat content at time t of R^m \ \beta [0, s] for fixed s and m = 2,3 in the limit t \downarrow 0, when \beta [0, s] is kept at temperature 1 for all t > 0 and R^m \ \beta [0, s] has initial temperature 0, and (ii) the inradius of T^m \ \beta [0, s] for m = 2,3,… in the limit s \rightarrow \infty. Key words and phrases. Laplacian, Brownian motion, Wiener sausage, heat content, inradius, spectrum