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 \beta with the length of the polymer n. We scale \beta_n := \beta n^{-\alpha} for alpha non-negative. This scaling sits in between the usual weak disorder (\beta = 0) and strong disorder regimes (\beta > 0). The fluctuation exponents zeta for the polymer endpoint and \chi for the free energy depend on \alpha in this regime, with \alpha = 0 corresponding to the usual polymer exponents \zeta = 2/3, \chi = 1/3 and \alpha >= 1/4 corresponding to the simple random walk exponents \zeta = 1/2, \chi = 0. For 0 < \alpha < 1/4 the exponents interpolate linearly between these two extremes. At \alpha = 1/4 we exactly identify the limiting distribution of the free energy and the end point of the polymer.Comment: 4 pages, 1 figure. Added a more detailed description of scaling results in the critical regime. Improved simulation results

Diffusions of Multiplicative Cascades

A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random variables on the vertices with independent increment processes satisfying certain moment assumptions. Our process has a Markov property: at any given time it is a cascade of the process at any earlier time by random variables that are independent of the past. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. We discuss applications of this process to models of tree polymers and one-dimensional random geometry.Comment: 30 pages; added section on Holder continuit

Bak-Sneppen backwards

We study the backwards Markov chain for the Bak–Sneppen model of biological evolution and derive its corresponding reversibility equations. We show that, in contrast to the forwards Markov chain, the dynamics of the backwards chain explicitly involve the stationary distribution of the model, and from this we derive a functional equation that the stationary distribution must satisfy. We use this functional equation to derive differential equations for the stationary distribution of Bak–Sneppen models in which all but one or all but two of the fitnesses are replaced at each step, subject to certain conditions on the relative locations of the replaced species. This gives a unified way of deriving Schlemm’s expressions for the stationary distributions of the isotropic four-species model, the isotropic five-species model, and the anisotropic three-species model

The intermediate disorder regime for directed polymers in dimension 1+1

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 β_n:=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