Interface theory and percolation

Abstract

This thesis is mainly concerned with percolation on general infinite graphs, as well as the approximation of conformal maps by square tilings, which are defined using electrical networks. The first chapter is concerned with the smoothness of the percolation density on various graphs. In particular, we prove that for Bernoulli percolation on Z d , d ≥ 2, the percolation density is an analytic function of the parameter in the supercritical interval (pc(Z d ), 1]. This answers a question of Kesten [1981]. The analogous result is also proved for the Boolean model of continuum percolation in R 2 , answering a question of Last et al. [2017]. In order to prove these results, we introduce the notion of interfaces, which is studied extensively in the current thesis. For dimensions d ≥ 3, we use renormalisation tecnhiques. Furthermore, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that pc < 1/2 for bond percolation on certain families of triangulations for which Benjamini & Schramm conjectured that pc ≤ 1/2 for site percolation. For the latter result, we use the well-known circle packing theorem of He and Schramm [1995], a discrete analogue of the Riemann mapping theorem. In Chapter 2, we continue the study of interfaces, and in particular, we consider the exponential growth rate br of the number of interfaces of a given size as a function of their surface-to-volume ratio r. We prove that the values of the percolation parameter p for which the interface size distribution has an exponential tail are uniquely determined by br by comparison with a dimension-independent function f(r) := (1+r) 1+r r r . We also point out a formula for translating any upper bound on the percolation threshold of a lattice G into a lower bound on the exponential growth rate of lattice animals a(G) and vice-versa. We exploit this in both directions. We obtain the rigorous lower bound pc(Z 3 ) > 0.2522 for 3-dimensional site percolation. We also improve on the best known asymptotic lower and upper bounds on a(Z d ) as d → ∞. We also prove that the rate of the exponential decay of the cluster size distribution, defined as c(p) := limn→∞ (Pp(|Co| = n))1/n, is a continuous function of p. The proof makes use of the Arzel`a-Ascoli theorem but otherwise boils down to elementary calculations. The analogous statement is also proved for the interface size distribution. For this we first establish that the rate of exponential decay is well-defined. In Chapter 3, we use interfaces to obtain upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. The results of this chapter are inspired by well-known conjectures of Benjamini and Schramm [1996b] for percolation on general graphs. We prove a conjecture by Benjamini and Schramm [1996b] stating that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2. We also make progress on a conjecture of Angel et al. [2018] that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. In the process, we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of [Lyons and Peres, 2016, Question 6.20]. Another topic of this thesis is the discrete approximation of conformal maps using another discrete analogue of the Riemann mapping theorem, namely the square tilings of Brooks et al. [1940]. This result is analogous to a well-known the orem of Rodin & Sullivan, previously conjectured by Thurston, which states that the circle packing of the intersection of a lattice with a simply connected planar domain Ω into the unit disc D converges to a Riemann map from Ω to D when the mesh size converges to 0. As a result, we obtain a new algorithm that allows us to numerically compute the Riemann map from any Jordan domain onto a square