TY - JOUR

T1 - Tightness and tails of the maximum in 3D Ising interfaces

AU - Gheissari, Reza

AU - Lubetzky, Eyal

N1 - Funding Information:
where the last equality is again by our weak convergence assumption. Together, this implies that, for every x ∈ R, we have Gk(x + ak) = G(x); having established this for every k ≥ 2, we find that G is max-stable, yet G is discrete, contradicting the fact that the only (nondegenerate) max-stable distributions are continuous ones, belonging to one of the three classes of extreme value distributions (see, e.g., [21], Theorem 1.3.1). □ Acknowledgments. We are grateful to an anonymous referee for valuable comments. E.L. was supported in part by NSF Grant DMS-1812095.
Publisher Copyright:
© 2021. Institute of Mathematical Statistics. All Rights Reserved.

PY - 2021/3

Y1 - 2021/3

N2 - Consider the 3D Ising model on a box of side length n with minus boundary conditions above the xy-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height Mn of this interface: for every β large, Mn/log n cβ in probability as n ∞ Here, we show that the laws of the centered maxima (Mn — (Formula presented)[Mn])n >1 are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to O(1) precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.

AB - Consider the 3D Ising model on a box of side length n with minus boundary conditions above the xy-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height Mn of this interface: for every β large, Mn/log n cβ in probability as n ∞ Here, we show that the laws of the centered maxima (Mn — (Formula presented)[Mn])n >1 are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to O(1) precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.

KW - 3D ising model

KW - low temperature interface

KW - maximum of random surface

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U2 - 10.1214/20-AOP1459

DO - 10.1214/20-AOP1459

M3 - Article

AN - SCOPUS:85120935386

SN - 0091-1798

VL - 49

SP - 732

EP - 792

JO - Annals of Probability

JF - Annals of Probability

IS - 2

ER -