We introduce a new disorder regime for directed polymers in dimension 1 + 1 that sits between the weak and strong disorder regimes. We call it the intermediate disorder regime. It is accessed by scaling the inverse temperature parameter β to zero as the polymer length n tends to infinity. The natural choice of scaling is βn := βn −1/4 . We show that the polymer measure under this scaling has previously unseen behavior. While the fluctuation exponents of the polymer endpoint and the log partition function are identical to those for simple random walk (ζ = 1/2, χ = 0), the fluctuations themselves are different. These fluctuations are still influenced by the random environment, and there is no self-averaging of the polymer measure. In particular, the random distribution of the polymer endpoint converges in law (under a diffusive scaling of space) to a random absolutely continuous measure on the real line. The randomness of the measure is inherited from a stationary process A β that has the recently discovered crossover distributions as its one-point marginals, which for large β become the GUE Tracy-Widom distribution. We also prove existence of a limiting law for the four-parameter field of polymer transition probabilities that can be described by the stochastic heat equation.In particular, in this weak noise limit, we obtain the convergence of the point-to-point free energy fluctuations to the GUE Tracy-Widom distribution. We emphasize that the scaling behaviour obtained is universal and does not depend on the law of the disorder.
Motivated by discrete directed polymers in one space and one time dimension,
we construct a continuum directed random polymer that is modeled by a
continuous path interacting with a space-time white noise. The strength of the
interaction is determined by an inverse temperature parameter beta, and for a
given beta and realization of the noise the path evolves in a Markovian way.
The transition probabilities are determined by solutions to the one-dimensional
stochastic heat equation. We show that for all beta > 0 and for almost all
realizations of the white noise the path measure has the same Holder continuity
and quadratic variation properties as Brownian motion, but that it is actually
singular with respect to the standard Wiener measure on C([0,1]).Comment: 21 page
We consider chordal SLE(kappa) curves for kappa > 4, where the intersection
of the curve with the boundary is a random fractal of almost sure Hausdorff
dimension min {2-8/kappa,1}. We study the random sets of points at which the
curve collides with the real line at a specified "angle" and compute an almost
sure dimension spectrum describing the metric size of these sets. We work with
the forward SLE flow and a key tool in the analysis is Girsanov's theorem,
which is used to study events on which moments concentrate. The two-point
correlation estimates are proved using the direct method.Comment: 21 page
We establish an upper bound on the asymptotic probability of an SLE(κ) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < κ < 8. As a consequence we are able to prove that the random set of points in R hit by the curve has Hausdorff dimension 2 − 8/κ, almost surely.2000 Mathematics Subject Classification. 60D05, 60K35, 28A80
We derive a number of estimates for the probability that a chordal SLE κ path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0 < κ < 8 andwhere a = 2/κ and C(x; rx) denotes the semicircle centred at x > 0 of radius rx, 0 < r ≤ 1/3, in the upper half plane. As an application of our results, for 0 < κ < 8, we derive an estimate for the diameter of a chordal SLE κ path in H between two real boundary points 0 and x > 0. For 4 < κ < 8, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE κ path in H from 0 to ∞.2000 Mathematics Subject Classification. 82B21, 60K35, 60G99, 60J65
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