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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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