arXiv subjects

Tom Alberts

Publications and source records attributed to Tom Alberts.

23 records · Page 2Linked to original sources

arXiv2010

Bridge Decomposition of Restriction Measures

Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the SLE(8/3) process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of SLE(8/3), and as a result can be generalized to the wider class of restriction measures. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of irreducible bridges in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros.

math.PR↗

arXiv2010

The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1

We introduce a new disorder regime for directed polymers with one space and one time dimension that is accessed by scaling the inverse temperature parameter βwith the length of the polymer n. We scale β_n := βn^{-α} for alpha non-negative. This scaling sits in between the usual weak disorder (β= 0) and strong disorder regimes (β> 0). The fluctuation exponents zeta for the polymer endpoint and χfor the free energy depend on αin this regime, with α= 0 corresponding to the usual polymer exponents ζ= 2/3, χ= 1/3 and α>= 1/4 corresponding to the simple random walk exponents ζ= 1/2, χ= 0. For 0 < α< 1/4 the exponents interpolate linearly between these two extremes. At α= 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.

cond-mat.stat-mech↗

arXiv2008

The Covariant Measure of SLE on the Boundary

We construct a natural measure mu supported on the intersection of a chordal SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The measure is a function of the SLE path in question. Assuming that boundary measures transform in a ``d-dimensional'' way (where d is the Hausdorff dimension of gamma intersected with R), we show that the measure we construct is (up to multiplicative constant) the unique measure-valued function of the SLE path that satisfies the Domain Markov property.

math.PR↗

arXiv2008

Intersection probabilities for a chordal SLE path and a semicircle

We derive a number of estimates for the probability that a chordal SLE path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0 0 of radius rx, 0 0. For 4<kappa<8, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE path in H from 0 to infinity.

math.PR↗

arXiv2007

Hausdorff dimension of the SLE curve intersected with the real line

We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a consequence we are able to prove that the SLE curve intersected with the real line has Hausdorff dimension 2-8/kappa, almost surely.

math.PR↗