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Integrals, partitions, and cellular automata. (English) Zbl 1095.60003
Let \(f:[0,1]\to[0,1]\) be the decreasing function that satisfies \(f(x)^a-f(x)^b=x^a-x^b\) for all \(x\in[0,1]\) and some fixed \(0<a<b\). The main result states that \[ \int_0^1\frac{-\log f(x)}{x}\;dx=\frac{\pi^2}{3ab}\,. \] The authors show how to evaluate other definite integrals using this result. If \(a\) is a positive integer and \(b=a+1\), the main result yields 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.
For a probabilistic application, let \(C_1,C_2,\ldots\) be independent events with probabilities \(\mathbb{P}_s(C_n)=1-(1-s)^n\) for some \(s\in(0,1)\). Define \(A_k\) to be the event that there is no sequence of \(k\geq1\) consecutive \(C_i\)’s that do not occur. In this case the main result yields \(-\log\mathbb{P}_s(A_k)\sim \pi^2/(3k(k+1)s)\) as \(s\to0\).

MSC:
60C05 Combinatorial probability
11P81 Elementary theory of partitions
05A17 Combinatorial aspects of partitions of integers
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
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