Extremes of the two-dimensional Gaussian free field with scale-dependent variance

Abstract

In this paper, we study a random field constructed from the twodimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen et al. (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.

Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

“…Hence, from the remark at the end of Lemma 3.1 in Arguin and Ouimet (2016), we know that Theorem 5.1 and Theorem 5.2 (in this paper) hold on V δ N ; the proof is in fact easier. Since A N,ρ ⊇ V δ N for N large enough, we have…”

“…Theorem 6.1 not only generalizes Theorem 2.1 in Arguin and Zindy (2015), but is also stronger because it tells us that including points arbitrarily close to ∂V N in the free energy has no impact on its limit, as long as we include the center of V N . We are able to prove Theorem 6.1 here because the asymptotics of |H N (γ)| were proved on V N in Arguin and Ouimet (2016).…”

“…In fact, from Lemma 3.4 and Lemma 3.5 in Arguin and Ouimet (2016), for any γ ∈ [0, γ ] and for any ε > 0, there exists a constant c = c(γ, ε, σ, λ) > 0 such that for N large enough, (5.8) and, when γ ∈ [0, γ ),…”

“…On the other hand, from Lemma A.2 in Arguin and Ouimet (2016) and the independence of the increments, we know that for any δ ∈ (0, 1/2] and j ∈ {1, 2, ..., m}, then for N large enough and all…”

“…Remark 6.9. It is expected that the results of Arguin and Ouimet (2016) on the first order asymptotics of the maximum and γ-high points can be extended to the more general case where the variance function σ in (4.1) is piecewise C 1 . Therefore, it is also expected that the results in the present article could be generalised just like Bovier and Kurkova (2004b) did when they generalized the results of the GREM to the CREM (the GREM with a continuum of hierarchies).…”