Learn More
This paper continues the analysis, begun in [4], of the asymptotic behavior of sums S, of certain dependent random variables which occur in statistical mechanics. These random variables are associated with the Curie-Weiss, or mean field model, a lattice model of ferromagnetism [1], [8; w [11; Ch. 61. We briefly indicate how the results of the present paper(More)
Let T (x; r) denote the occupation measure of the disc of radius r centered at x by planar Brownian motion run till time 1. We prove that sup jxjj1 T (x; r)=(r 2 j log rj 2) ! 2 a.s. as r ! 0, thus solving a problem posed by Perkins and Taylor (1987). Furthermore, we show that for any a < 2, the Hausdorr dimension of the set of \perfectly thick points" x(More)
Let X = {X(t), t ∈ R+} be a real-valued symmetric Lévy process with continuous local times {L x t , (t, x) ∈ R+ × R} and characteristic function Ee iλX(t) = e −tψ(λ). Let σ 2 0 (x − y) = 4 π ∞ 0 sin 2 (λ(x − y)/2) ψ(λ) dλ. If σ 2 0 (h) is concave, and satisfies some additional very weak regularity conditions, then for any p ≥ 1, and all t ∈ R+, lim h↓0 b a(More)
Let Tn(x) denote the time of first visit of a point x on the lattice torus Zn = Z /nZ by the simple random walk. The size of the set of α, n-late points Ln(α) = {x ∈ Zn :Tn(x) ≥ α 4 π (n logn) } is approximately n, for α ∈ (0,1) [Ln(α) is empty if α> 1 and n is large enough]. These sets have interesting clustering and fractal properties: we show that for β(More)
Let T (x, ε) denote the first hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T2. We prove that supx∈T2 T (x, ε)/| log ε|2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a(More)
Let x r denote the total occupation measure of the ball of radius r centered at x for Brownian motion in 3. We prove that sup x ≤1 x r / r2 log r → 16/π2 a.s. as r → 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a ∈ 0 16/π2 , the Hausdorff dimension of the set of “thick points” x for which lim supr→0 x r / r2 log r = a is almost(More)
We show that the renormalized self-intersection local time γt(x) for both the Brownian motion and symmetric stable process in R is differentiable in the spatial variable and that γ′ t(0) can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a(More)