2023/02/03 by Philip Easo, Tom Hutchcroft, Easo, Philip +3 · 1 citation
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2302.01509
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 and 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.