Dan Romik

Projecting the surface measure of the sphere of ℓpn

Abstract We prove that the total variation distance between the cone measure and surface measure on the sphere of lpn is bounded by a constant times 1/ n . This is used to give a new proof of the fact that the coordinates of a random vector on the lpn sphere are approximately independent with density proportional to exp(−|t|p), a unification and generalization of two theorems of Diaconis and Freedman. Finally, we show in contrast that a projection of the surface measure of the lpn sphere onto a random k-dimensional subspace is “close” to the k-dimensional Gaussian measure.

Integrals, partitions, and cellular automata

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.

Shortest Paths in the Tower of Hanoi Graph and Finite Automata

We present efficient algorithms for constructing a shortest path between two configurations in the Tower of Hanoi graph and for computing the length of the shortest path. The key element is a finite-state machine which decides, after examining on the average only a small number of the largest discs (asymptotically,

Random sorting networks

\frac{63}{38} \approx 1.66

Integrals, partitions and MacMahon's Theorem

), whether the largest disc will be moved once or twice. This solves a problem raised by Andreas Hinz and results in a better understanding of how the shortest path is determined. Our algorithm for computing the length of the shortest path is typically about twice as fast as the existing algorithm. We also use our results to give a new derivation of the average distance

Arctic circles, domino tilings and square Young tableaux

\frac{466}{885}

New enumeration formulas for alternating sign matrices and square ice partition functions

between two random points on the Sierpin´ski gasket of unit side.

The oriented swap process

Abstract 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 space–time 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 Holder-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.

The dynamics of Pythagorean Triples

In two previous papers, the study of partitions with short sequences has been developed both for its intrinsic interest and for a variety of applications. The object of this paper is to extend that study in various ways. First, the relationship of partitions with no consecutive integers to a theorem of MacMahon and mock theta functions is explored independently. Secondly, we derive in a succinct manner a relevant definite integral related to the asymptotic enumeration of partitions with short sequences. Finally, we provide the generating function for partitions with no sequences of length K and part exceeding N.

The number of guillotine partitions in d dimensions

The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the “arctic circle” inscribed within the diamond. A similar arctic circle phenomenon has been observed in the limiting behavior of random square Young tableaux. In this paper, we show that random domino tilings of the Aztec diamond are asymptotically related to random square Young tableaux in a more refined sense that looks also at the behavior inside the arctic circle. This is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the Aztec diamond that uses the large-deviation techniques developed for the square Young tableaux problem in a previous paper by Pittel and the author. The solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square Young tableaux by Pittel and the author. The analytic techniques used to solve the variational problem provide a systematic, guess-free approach for solving problems of this type which have appeared in a number of related combinatorial probability models.