1202.4398 (Tom Alberts et al.)

The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1    [PDF]

Tom Alberts, Konstantin Khanin, Jeremy Quastel

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 beta to zero as the polymer length n tends to infinity. The natural
choice of scaling is beta*n^{-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 (zeta = 1/2, chi = 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_{beta}
that has the recently discovered crossover distributions as its one-point
marginals. We also prove existence of a limiting law for the four-parameter
field of polymer transition probabilities. The limit can be described by the
stochastic heat equation.

View original: http://arxiv.org/abs/1202.4398