Short- and long-time path tightness of the continuum directed random polymer

TL;DR

This paper studies the behavior of the continuum directed random polymer model at short and long times, showing convergence to Brownian bridges and geodesics of the directed landscape, with tightness and fluctuation results.

Contribution

It provides the first rigorous analysis of the path tightness and fluctuation exponents for the continuum directed random polymer at different time scales.

Findings

Short-time law converges to Brownian bridge.

Long-time fluctuations follow a 2/3 exponent.

Path measures are tight under diffusive and 2/3 scaling.

Abstract

We consider the point-to-point continuum directed random polymer ($\mathsf{CDRP}$) model that arises as a scaling limit from $1+1$ dimensional directed polymers in the intermediate disorder regime. We show that the annealed law of a point-to-point $\mathsf{CDRP}$ of length $t$ converges to the Brownian bridge under diffusive scaling when $t \downarrow 0$. In case that $t$ is large, we show that the transversal fluctuations of point-to-point $\mathsf{CDRP}$ are governed by the $2/3$ exponent. More precisely, as $t$ tends to infinity, we prove tightness of the annealed path measures of point-to-point $\mathsf{CDRP}$ of length $t$ upon scaling the length by $t$ and fluctuations of paths by $t^{2/3}$. The $2/3$ exponent is tight such that the one-point distribution of the rescaled paths converges to the geodesics of the directed landscape. This point-wise convergence can be enhanced to process-level modulo a conjecture. Our short and long-time tightness results also extend to point-to-line $\mathsf{CDRP}$. In the course of proving our main results, we establish quantitative versions of quenched modulus of continuity estimates for long-time $\mathsf{CDRP}$ which are of independent interest.