Integrals, partitions, and cellular automata

By Alexander E. Holroyd, Thomas M. Liggett, Dan Romik. 2003

73 citations recorded by the source corpus.

We prove that ∫ 1 0 -log f(x)/x dx = π 2 /3ab, 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2246774eac66a74c20db0ca2bbd6b116c6c1154a

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

The sharp threshold for bootstrap percolation in all dimensions (2010, background): “In particular, λ ( d, r ) ∞ for every d (cid:2) r (cid:2) 2, λ (2 , 2) = π 2 18 (see [31] and also [33]), λ ( d, 2) = d − 1 2 + o (1), and as d → ∞ (see [6]).”
An extension of the Hardy-Ramanujan circle method and applications to partitions without sequences (2011, methodology): “In particular, we will use the important example of “partitions without sequences” to illustrate the application of our general results throughout the rest of the paper, but we emphasize that our approach of identifying principal part integrals in Sections 3 and 4 is widely applicable to other…”