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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: December 15, 2003
MathSciNet review: 2052953
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that

\begin{displaymath}\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab},\end{displaymath}

where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ 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: http://dx.doi.org/10.1090/S0002-9947-03-03417-2
PII: S 0002-9947(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