# High-dimensional near-critical percolation and the torus plateau

@inproceedings{Hutchcroft2021HighdimensionalNP, title={High-dimensional near-critical percolation and the torus plateau}, author={Tom Hutchcroft and Emmanuel Michta and Gordon Slade}, year={2021} }

We consider percolation on Z and on the d-dimensional discrete torus, in dimensions d ≥ 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Z, we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy Ppc−ε(0↔ x) ≤ C ‖x‖d−2 e−cε ‖x‖ and c r2 e−Cε r ≤ Ppc−ε ( 0↔ ∂[−r, r] ) ≤ C r2 e−cε 1/2r. Using this, we prove that…

## 8 Citations

Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

- Computer Science
- 2021

This study provides sharp estimates for several quantities of interest at the critical probability pc, including the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at “mesoscopic distance” from the boundary of half-spaces.

The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions

- Mathematics
- 2020

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice Z in dimensions d > 4, in the vicinity of the critical point,…

Sharpness of Bernoulli percolation via couplings

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- 2022

In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice Z). We prove the following estimate, where θn(p) is the probability…

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- 2021

We establish the mean-field bounds γ ≥ 1, δ ≥ 2 and 4 ≥ 2 on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously…

The scaling limit of the weakly self-avoiding walk on a high-dimensional torus

- Mathematics
- 2021

We prove that the scaling limit of the weakly self-avoiding walk on a d-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is o(V ) where V is the…

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- Mathematics, Physics
- 2022

We study unwrapped two-point functions for the Ising model, the selfavoiding walk and a random-length loop-erased random walk on high-dimensional lattices with periodic boundary conditions. While the…

Ising Model with Curie–Weiss Perturbation

- MathematicsJournal of Statistical Physics
- 2022

. Consider the nearest-neighbor Ising model on Λ n := [ − n, n ] d ∩ Z d at inverse temperature β ≥ 0 with free boundary conditions, and let Y n ( σ ) := P u ∈ Λ n σ u be its total magnetization. Let…

Asymptotic behaviour of the lattice Green function

- Mathematics
- 2021

The lattice Green function, i.e., the resolvent of the discrete Laplace operator, is fundamental in probability theory and mathematical physics. We derive its long-distance behaviour via a detailed…

## References

SHOWING 1-10 OF 52 REFERENCES

Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time

- Mathematics
- 2009

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich…

Random Graph Asymptotics on High-Dimensional Tori

- Mathematics
- 2005

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently…

Mean-field behavior for nearest-neighbor percolation in $d>10$

- Mathematics
- 2015

We prove that nearest-neighbor percolation in dimensions d ≥ 11 displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle…

Gaussian fluctuations of connectivities in the subcritical regime of percolation

- Mathematics
- 1991

SummaryWe consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp<pc: (1) The leading power-law correction to exponential decay of the connectivity…

Tree graph inequalities and critical behavior in percolation models

- Mathematics
- 1984

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent γ associated with the expected…

Mean-field critical behaviour for percolation in high dimensions

- Mathematics
- 1990

AbstractThe triangle condition for percolation states that
$$\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} $$
is finite at the critical point, where τ(x, y) is the probability that the…

The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions

- Mathematics
- 2020

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice Z in dimensions d > 4, in the vicinity of the critical point,…

Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

- Mathematics
- 2000

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical…

Cycle structure of percolation on high-dimensional tori

- Mathematics
- 2011

Abstract In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The…