Double-exponential susceptibility growth in Dyson's hierarchical model with |x − y|⁻² interaction

Creators: Easo, Philip; Hutchcroft, Tom¹; Kurrek, Jana

1. California Institute of Technology

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 β_c < ∞ if and only if α < d. We study the model in the regime α ≥ d, in which β_c = ∞, and prove that the susceptibility χ(β) (i.e., the expected volume of the cluster at the origin) satisfies χ(β) = β(d/1−d)^(−o(1)) as β↑∞ if α > d and χ(β) = e^(e^e(Θ(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.

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Acknowledgement

This work was carried out as part of Caltech’s Summer Undergraduate Research Fellowship (SURF) program 2022, during which J.K. was mentored by P.E. and T.H. During the research, J.K. was also supported by an NSERC USRA. We thank Louigi Addario-Berry and Johannes Bäumler for helpful comments on a draft.

Contributions

Philip Easo: Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (supporting); Writing – original draft (equal); Writing – review & editing (supporting). Tom Hutchcroft: Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (lead). Jana Kurrek: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (supporting).

Data Availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Conflict of Interest

The authors have no conflicts to disclose.

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