Integrals, partitions, and cellular automata

Authors:
Alexander E. Holroyd, Thomas M. Liggett and Dan Romik

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3349-3368

MSC (2000):
Primary 26A06; Secondary 05A17, 60C05, 60K35

DOI:
https://doi.org/10.1090/S0002-9947-03-03417-2

Published electronically:
December 15, 2003

MathSciNet review:
2052953

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that

where is the decreasing function that satisfies , for . When is an integer and we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

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Additional Information

**Alexander E. Holroyd**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Email:
holroyd@math.ubc.ca

**Thomas M. Liggett**

Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, Califonia 90095-1555

Email:
tml@math.ucla.edu

**Dan Romik**

Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

Email:
romik@wisdom.weizmann.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-03-03417-2

Keywords:
Definite integral,
partition asymptotics,
partition identity,
combinatorial probability,
threshold growth model,
bootstrap percolation,
cellular automaton

Received by editor(s):
February 17, 2003

Received by editor(s) in revised form:
May 6, 2003

Published electronically:
December 15, 2003

Additional Notes:
The first author’s research was funded in part by NSF Grant DMS–0072398.

The second author’s research was funded in part by NSF Grant DMS-00-70465.

Article copyright:
© Copyright 2003
American Mathematical Society