Statistics and Data Science Seminar
Mike Cranston
UCI
Overlaps and Pathwise Localization in the Anderson Polymer Model
Abstract: We consider large time behavior of typical paths under the Anderson polymer measure. If $P^x_\kappa$ is the measure induced by rate $\kappa,$ simple, symmetric random walk on $\mathbb{Z}^d$ started at $x,$ this measure is defined as
\[d\mu^x_{\kappa,\beta,T}T(X)={Z_{\kappa,\beta,T}}^{-1} \exp\left\{\beta\int_0^T dW_{X(s)}(s)\right\}dP^x_\kappa(X)\]
where $\{W_x:x\in \mathbb{Z}^d\}$ is a field of $iid$ standard, one-dimensional Brownian motions, $\beta>0, \kappa>0$ and
$Z_{\kappa,\beta,t}(x)$ the normalizing constant.
We establish that the polymer measure gives a macroscopic mass to a small neighborhood of a typical path as $T \to \infty$, for parameter values outside the perturbative regime of the random walk, giving a pathwise
approach to polymer localization, in contrast with existing results. The localization becomes complete as $\frac{\beta^2}{\kappa}\to\infty$ in the sense that the mass grows to 1.
The proof makes use of the overlap between two independent samples drawn under the Gibbs measure $\mu^x_{\kappa,\beta,T}$,
which can be estimated by the integration by parts formula for the Gaussian environment.
Conditioning this measure on the number of jumps, we obtain a canonical measure which already shows scaling
properties, thermodynamic limits, and decoupling of the parameters. This talk is based on joint work with Francis Comets.
Wednesday November 2, 2016 at 4:00 PM in SEO 636