The critical length for growing a droplet / Paul Balister, Béla Bollobás, Robert Morris, Paul Smith.

Format:

Book

Author/Creator:

Balister, Paul, author.

Bollobás, Béla, author.

Morris, Robert, author.

Smith, Paul, author.

Series:

Memoirs of the American Mathematical Society ; v. 1571.

Memoirs of the American Mathematical Society, 0065-9266 ; v. 1571

Language:

English

Subjects (All):

Probabilities.

Mathematical physics.

Combinatorial analysis.

probability.

Physical Description:

v, 180 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2025.

Summary:

In many interacting particle systems, relaxation to equilibrium is thought to occur via the growth of 'droplets', and it is a question of fundamental importance to determine the critical length at which such droplets appear. In this paper we construct a mechanism for the growth of droplets in an arbitrary finite-range monotone cellular automaton on a d-dimensional lattice. Our main application is an upper bound on the critical probability for percolation that is sharp up to a constant factor in the exponent. Our method also provides several crucial tools that we expect to have applications to other interacting particle systems, such as kinetically constrained spin models on Zd.

Contents:

Chapter 1. Introduction

Chapter 2. The resistance of an update family

Chapter 3. The main theorem and an outline of the proof

Chapter 4. Locally inherited resistance

Chapter 5. Finding a rational direction in which to grow

Chapter 6. Construction of the quasistable set

Chapter 7. The resistance of induced update families

Chapter 8. Polytopes

Chapter 9. The bootstrap process in a polytope

Chapter 10. Interiors, extensions, buffers, and growth sequences

Chapter 11. Deterministic growth of droplets

Chapter 12. The proof of Theorem 3.1

Appendix A. Properties of canonical polytopes

Appendix B. The distance between faces of a polytope

Appendix C. Interiors and extensions

Appendix D. Some technical details from Chapters 11 and 12

Appendix E. Perfectly covering a polytope with smaller polytopes

Bibliography.

Notes:

Number 1571 (sixth of 6 numbers)

Includes bibliographical references (pages 177-180).

ISBN:

1470474875

9781470474874

OCLC:

1527985571