Double-exponential susceptibility growth in Dyson’s hierarchical model with |xy|−2 interaction

Author:

Easo Philip1ORCID,Hutchcroft Tom1ORCID,Kurrek Jana2ORCID

Affiliation:

1. The Division of Physics, Mathematics and Astronomy, California Institute of Technology 1 , Pasadena, California 91125, USA

2. Department of Mathematics and Statistics, McGill University 2 , Montreal, Quebec H3A 0G4, Canada

Abstract

We study long-range percolation on the d-dimensional hierarchical lattice, in which each possible edge {x, y} is included independently at random with inclusion probability 1 − exp(−β ‖x − y‖−d−α), where α > 0 is fixed and β ≥ 0 is a parameter. This model is known to have a phase transition at some βc < ∞ if and only if α < d. We study the model in the regime α ≥ d, in which βc = ∞, and prove that the susceptibility χ(β) (i.e., the expected volume of the cluster at the origin) satisfies χ(β)=βdα−d−o(1) as β↑∞ if α > d and χ(β)=eeΘ(β) as β↑∞ if α = d. This resolves a problem raised by Georgakopoulos and Haslegrave (2020), who showed that χ(β) grows between exponentially and double-exponentially when α = d. Our results imply that analogous results hold for a number of related models including Dyson’s hierarchical Ising model, for which the double-exponential susceptibility growth we establish appears to be a new phenomenon even at the heuristic level.

Funder

Caltech SURF

NSERC USRA

Publisher

AIP Publishing

Reference25 articles.

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