Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates

TL;DR

This paper investigates the scaling limits and law of large numbers for weakly pinned Gaussian random fields in higher dimensions, especially when multiple limit candidates exist, extending previous one-dimensional results.

Contribution

It extends the understanding of Gaussian random fields by analyzing higher-dimensional cases with multiple potential limits under weak pinning conditions.

Findings

Established scaling limits for d ≥ 3

Proved law of large numbers in critical pinning scenarios

Identified conditions for multiple limit candidates

Abstract

We study the scaling limit and prove the law of large numbers for weakly pinned Gaussian random fields under the critical situation that two possible candidates of the limits exist at the level of large deviation principle. This paper extends the results of [3], [7] for one dimensional fields to higher dimensions: d \geq 3, at least if the strength of pinning is sufficiently large.