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Expansion of Percolation Critical Points for Hamming Graphs

Published online by Cambridge University Press:  05 August 2019

Lorenzo Federico
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
Remco Van Der Hofstad
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mails: r.w.v.d.hofstad@tue.nl, w.j.t.hulshof@tue.nl)
Frank Den Hollander
Affiliation:
Mathematisch Instituut, Leiden University, 2333 CA Leiden, The Netherlands (e-mail: denholla@math.leidenuniv.nl)
Tim Hulshof
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mails: r.w.v.d.hofstad@tue.nl, w.j.t.hulshof@tue.nl)

Abstract

The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ be the number of vertices of H(d, n). Let $p_c^{(d)}$ be the critical point for bond percolation on H(d, n). We show that, for $d \in \mathbb{N}$ fixed and $n \to \infty$,

$$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$
which extends the asymptotics found in [10] by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} = O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös–Rényi random graph.

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© Cambridge University Press 2019 

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