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) … Show more
“…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…”
Section: Proofs Of the Main Resultsmentioning
confidence: 62%
“…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).…”
Section: New Resultsmentioning
confidence: 97%
“…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 ),…”
Section: Previous Resultsmentioning
confidence: 99%
“…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…”
Section: Proofs Of the Main Resultsmentioning
confidence: 99%
“…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).…”
We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V N and adapt a technique of Bovier and Kurkova (2004b) to determine the limiting two-overlap distribution. The adaptation was already successfully applied in the simpler case of Arguin and Zindy (2015), where the limiting free energy was computed for the field with two levels (in the center of V N ) and the limiting twooverlap distribution was determined in the homogeneous case. Our results agree with the analogous quantities for the Generalized Random Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova (2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra identities hold exactly in the limit. As a corollary, the limiting array of overlaps is ultrametric and the limiting Gibbs measure has the same law as a Ruelle probability cascade.
“…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…”
Section: Proofs Of the Main Resultsmentioning
confidence: 62%
“…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).…”
Section: New Resultsmentioning
confidence: 97%
“…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 ),…”
Section: Previous Resultsmentioning
confidence: 99%
“…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…”
Section: Proofs Of the Main Resultsmentioning
confidence: 99%
“…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).…”
We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V N and adapt a technique of Bovier and Kurkova (2004b) to determine the limiting two-overlap distribution. The adaptation was already successfully applied in the simpler case of Arguin and Zindy (2015), where the limiting free energy was computed for the field with two levels (in the center of V N ) and the limiting twooverlap distribution was determined in the homogeneous case. Our results agree with the analogous quantities for the Generalized Random Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova (2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra identities hold exactly in the limit. As a corollary, the limiting array of overlaps is ultrametric and the limiting Gibbs measure has the same law as a Ruelle probability cascade.
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