Analyticity Results in Bernoulli Percolation / Agelos Georgakopoulos and Christoforos Panagiotis.

Format:

Book

Author/Creator:

Georgakopoulos, Agelos, author.

Panagiotis, Christoforos, author.

Series:

Memoirs of the American Mathematical Society ; Volume 288.

Memoirs of the American Mathematical Society Series ; Volume 288

Language:

English

Subjects (All):

Percolation (Statistical physics).

Combinatorial enumeration problems.

Physical Description:

1 online resource (114 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We prove that for Bernoulli percolation on Zd, d (greater than or equal to) 2, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive shortor long-range models, and that pbond c (less than) 1/2 for certain families of triangulations for which Benjamini & Schramm conjectured that psite c (less than or equal to) 1/2"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Background and motivation

1.2. Summary of results

1.3. Proof ideas

Chapter 2. The setup

2.1. Nearest-neighbour models

2.2. Long-range models

Chapter 3. Definitions and preliminaries

3.1. Graph theoretic definitions

3.2. Exponential tail of the subcritical cluster size distribution: The Aizenman-Newman-Barsky property

3.3. The BK inequality

3.4. Partitions of integers

Chapter 4. The basic technique

4.1. Nearest-neighbour models

4.2. Long-range models

4.3. Analyticity of susceptibility in the subcritical regime

Chapter 5. Analyticity for non-amenable graphs

Chapter 6. Trees

6.1. Regular trees

6.2. Galton-Watson trees

6.3. A tree with percolation density nowhere analytic

Chapter 7. Analyticity above the threshold for planar lattices

7.1. Preliminaries on planar quasi-transitive lattices

7.2. Main result

7.3. Site percolation

Chapter 8. Analyticity of percolation density in all dimensions

8.1. Setting up the renormalisation

8.2. Separating components

8.3. Expressing the percolation density using separating components

8.4. Expanding the percolation density as an infinite sum of polynomials

8.5. Exponential tail of a certain cutset

8.6. Analyticity of k-point function

Chapter 9. Continuum percolation

Chapter 10. Finitely presented groups

10.1. The setup and notation

10.2. A connectedness concept

10.3. Interfaces

10.4. Properties of interfaces

10.5. Using interfaces to prove analyticity

10.6. Extending to site percolation

Chapter 11. Triangulations

11.1. Overview

11.2. Proofs

11.3. Site percolation

Chapter 12. Alternating signs of Taylor coefficients

Chapter 13. The negative percolation threshold

Appendix A. On the number of lattice animals of a given size.

Appendix B. Complex analysis basics

Bibliography

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Georgakopoulos, Agelos Analyticity Results in Bernoulli Percolation

ISBN:

9781470475734

1470475731