A stationary model of non-intersecting directed polymers

TL;DR

This paper extends the stationary measure results of the directed polymer model from a single polymer to multiple non-intersecting polymers, revealing their asymptotic behavior and connections to Brownian motions and semi-discrete polymers.

Contribution

It introduces a stationary measure for multiple non-intersecting directed polymers and links it to non-intersecting Brownian motions and semi-discrete polymer models.

Findings

The ratio of partition functions converges to an explicit functional of Brownian motions.

The stationary measure of the multilayer stochastic heat equation is given by independent Brownian motions.

Explicit formulas are derived for the case of two polymers.

Abstract

We consider the partition function $Z_{\ell}(\vec x,0\vert \vec y,t)$ of $\ell$ non-intersecting continuous directed polymers of length $t$ in dimension $1+1$, in a white noise environment, starting from positions $\vec x$ and terminating at positions $\vec y$. When $\ell=1$, it is well known that for fixed $x$, the field $\log Z_1(x,0\vert y,t)$ solves the Kardar-Parisi-Zhang equation and admits the Brownian motion as a stationary measure. In particular, as $t$ goes to infinity, $Z_1(x,0\vert y,t)/Z_1(x,0\vert 0,t) $ converges to the exponential of a Brownian motion $B(y)$. In this article, we show an analogue of this result for any $\ell$. We show that $Z_{\ell}(\vec x,0\vert \vec y,t)/Z_{\ell}(\vec x,0\vert \vec 0,t) $ converges as $t$ goes to infinity to an explicit functional $Z_{\ell}^{\rm stat}(\vec y)$ of $\ell$ independent Brownian motions. This functional $Z_{\ell}^{\rm stat}(\vec y)$ admits a simple description as the partition sum for $\ell$ non-intersecting semi-discrete polymers on $\ell$ lines. We discuss applications to the endpoints and midpoints distribution for long non-crossing polymers and derive explicit formulas in the case of two polymers. To obtain these results, we show that the stationary measure of the O'Connell-Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric RSK correspondence found in arXiv:1110.3489. .