A sorting network is a shortest path from 12 · · · n to n · · · 21 in the Cayley graph of S n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the spacetime process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux. τ s 1 τ s 2 · · · τ s N = ρ

In a surprising sequence of developments, the longest increasing subsequence problem, originally mentioned as merely a curious example in a 1961 paper, has proven to have deep connections to many seemingly unrelated branches of mathematics, such as random permutations, random matrices, Young tableaux, and the corner growth model. The detailed and playful study of these connections makes this book suitable as a starting point for a wider exploration of elegant mathematical ideas that are of interest to every mathematician and to many computer scientists, physicists and statisticians. The specific topics covered are the Vershik-Kerov–Logan-Shepp limit shape theorem, the Baik–Deift–Johansson theorem, the Tracy–Widom distribution, and the corner growth process. This exciting body of work, encompassing important advances in probability and combinatorics over the last forty years, is made accessible to a general graduate-level audience for the first time in a highly polished presentation.

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Abstract. We prove thatwhere f (x) is the decreasing function that satisfiesWhen 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.