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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)

- AMIR DEMBO, YUVAL PERESy, +4 authors OFER ZEITOUNI
- 1999

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)

- Yuval Peres, Jay Rosen
- 2005

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)

- Amir Dembo, Yuval Peres, Jay Rosen
- 1998

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)

- Xia Chen, Jay Rosen
- 2005

We study large deviations for intersection local times of p independent d-dimensional symmetric stable processes of index β, under the condition p(d − β) < d. Our approach is based on FeynmanKac type large deviations, moment computations and some techniques from probability in Banach spaces.

- Jay Rosen
- 2007

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)

This research was supported, in part, by grants from the National Science Foundation, the Guggenheim Foundation, PSC-CUNY and an Scholar Incentive Award from The City College of CUNY. M. B. Marcus is grateful as well to Université Louis Pasteur and C.N.R.S., Strasbourg and the Statistical Laboratory and Clare Hall, Cambridge University for the support and… (More)

- Richard Bass, Xia Chen, Jay Rosen
- 2007

We study functionals of the form ζt = ∫ t