The endpoint distribution of directed polymers

Abstract

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called "partitioned subprobability measure", to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability. Contents2010 Mathematics Subject Classification. 60K37, 82B26, 82B44, 82D60.

Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder

“…The latter statement was later strengthened in [V07] for heavy-tailed environments (i.e., when the logarithmic moment generating function is infinity). Very recently, substantial progress was made when the latter statement was shown to be true in [BC16] for polymers in the lattice even with finite exponential moments. We also remark that localization properties for polymers in the lattice setting can be efficiently studied by using the method of fractional moments introduced in ([CSY03]).…”

“…Comparison with the earlier approach. As mentioned earlier, we have drawn inspiration from the techniques recently employed in [BC16] for directed polymers which also followed the program in [MV14] for constructing a metric on the compactification in a lattice setting. It was also shown ([BC16, Proposition A.3]) that the metric therein produces the same topology as [MV14] when the latter structure is adapted to the lattice setting.…”

“…It was also shown ([BC16, Proposition A.3]) that the metric therein produces the same topology as [MV14] when the latter structure is adapted to the lattice setting. However, the metric in [BC16] is structurally quite different † † For example, let µn be the Gaussian mixture 1 3 N (n, 1) + 1 3 N (n 2 , 1) + 1 3 N (0, n). Then the limiting object for µn is the collection ξ = { α1, α1} ∈ X , where α1 is the equivalence class of a Gaussian with variance 1 and weight 1 3 .…”

“…We will now conclude the proof of Theorem 1.1 in this section. Given the results of Section 3 and 2, the arguments appearing in this part will closely follow the approach of [BC16] adapted to our setting modulo slight modifications. In order to keep the present material self-contained, we will spell out the technical details.…”

The intermediate disorder regime for Brownian directed polymers in Poisson environment

“…For a preliminary step, one can define a favorite endpoint for the polymer on [0, t] in a similar way (see [13,Remark 2.3.1]). A complete understanding of the endpoint distribution has been recently achieved [4] in the discrete case.…”