Confinement of Brownian polymers under geometric area tilts

Abstract

We consider tightness for families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of 2 + 1 discrete Solid-On-Solid random interfaces above a hard wall. arXiv:1809.03209v1 [math.PR]

Tightness and Line Ensembles for Brownian Polymers Under Geometric Area Tilts

“…Our generalization [3] to measures with geometric area tilts reads as: The estimate above holds as well for the zero boundary distribution µ 0,k,γ n;T .…”

“…Our main motivation comes from a desire to understand and model limiting properties of microscopic large level lines of the 2 + 1 SOS (solid-on-solid) random interfaces conditioned to stay above a flat wall. We refer to [3] for a careful exposition of this connection with references to earlier works.…”

“…The Brownian notation and the models of Brownian polymers under geometric area tilts are introduced in Subsections 1.3-1.5. The main confinement results from [3] which enables a uniform control over one-point distributions of the top path is recalled in Subsection 1.6. Our main results here which are formulated in Subsection 1.7 yield full tightness of the ensembles we consider, both with free and zero boundary conditions, and, furthermore, we establish Brownian-Gibbs property of limiting infinite dimensional ensembles.…”

“…The corresponding partition function is denoted Z η,ν n; ,r (h − , h + , ρ). Below we shall tacitly assume that boundary conditions ν, η are chosen in such a way that the corresponding polymer measures are well defined, this is justified in all the relevant cases in Appendix A of [3].…”

“…Informally, ξ ,r α (h) is the minimal amount to lift ϕ α so that it will stay above h. We think of ξ ,r α (h) in terms of the curved maximum of h on [ , r]. Then, Theorem 1.1 is an immediate consequence of the following result [3]:…”