Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Extremal process in the weakly correlated regime

Abstract

We prove convergence of the full extremal process of the scale-inhomogeneous discrete Gaussian free field in dimension two in the weak correlation regime. The scale-inhomogeneous discrete Gaussian free field is obtained from the 2d discrete Gaussian free field by modifying the variance through a function I : [0, 1] → [0, 1]. The full extremal process converges to a cluster Cox process. The random intensity of the Cox process depends on I (0) through a random measure Y and on I (1) through a constant β. We show that, in law, the random measure, Y , is equal to the Liouville Quantum Gravity measure at sub-critical temperature α = 2σ(0). The cluster process, which only depends on I (1), can be described as atoms of a standard 2d discrete Gaussian free field conditioned to be unusually high.

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Extremal process in the weakly correlated regime

“…Under Assumption 1 we proved in [30,31], building on work by Arguin and Ouimet [7], the subleading order correction, tightness and convergence of the appropriately centred maximum. More explicitely, there exists a constant, β = β(σ(1)), which depends only on the final variance σ(1), and a random variable, Y = Y(σ(0)), depending only on the initial variance σ(0), such that, for any z ∈ R,…”

“…As depicted in [9,14], the fact that the limiting point process takes the particular form of a generalized Poisson point process, is a consequence of a superposition property, which is due to its Gaussian nature along with certain properties of the field such as the separation of local maxima [31] and tightness of extreme level sets. The main ingredient we need, in order to apply the machinery from [9] to obtain the distributional invariance and thus Poisson limit laws, is tightness of the point processes, which is a consequence of the following proposition and previous results in [31]. For y ∈ R, we denote by…”

“…We therefore omit details here. It essentially uses convergence of the maximum obtained in [31] together with expontential bounds on level sets, see Proposition 2.1.…”

“…Moreover, {Φ M,v v } v∈V N is a centred Gaussian field. Thus, we can rerun the proof of [31,Proposition 4.2], where the constant on the right of [31, (4.13)] may now depend on δ. This concludes the proof of Proposition 3.3.…”

“…N(0, log N). Moreover, we proved in [31,Theorem 2.2] that under Assumption 1, points whose height is close to the maximum are either O(N) apart or within distance O(1). In particular, there is a constant c > 0, such that…”