arXiv subjects

Dan Romik

Publications and source records attributed to Dan Romik.

32 records · Page 2Linked to original sources

arXiv2009

A formula for the doubly-refined enumeration of alternating sign matrices

Zeilberger proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey, for the number of alternating sign matrices with given top row. Stroganov proved an explicit formula for the number of alternating sign matrices with given top and bottom rows. Fischer and Romik considered a different kind of "doubly-refined enumeration" where one counts alternating sign matrices with given top two rows, and obtained partial results on this enumeration. In this paper we continue the study of the doubly-refined enumeration with respect to the top two rows, and use Stroganov's formula to prove an explicit formula for these doubly-refined enumeration numbers.

math.CO↗

arXiv2009

More refined enumerations of alternating sign matrices

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general "d-refined" enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_{n,i,j} that enumerate such matrices. We give a conjectural explicit formula for A_{n,i,j} and formulate several other conjectures about the sufficiency of the linear equations to determine the A_{n,i,j}'s and about an extension of the linear equations to the general d-refined enumerations.

math.CO↗

arXiv2009

Phase Transitions in Gravitational Allocation

Given a Poisson point process of unit masses (``stars'') in dimension d>=3, Newtonian gravity partitions space into domains of attraction (cells) of equal volume. In earlier work, we showed the diameters of these cells have exponential tails. Here we analyze the quantitative geometry of the cells and show that their large deviations occur at the stretched-exponential scale. More precisely, the probability that mass exp(-R^gamma) in a cell travels distance R decays like exp(-R^f_d(gamma)) where we identify the functions f_d exactly. These functions are piecewise smooth and the discontinuities of f_d' represent phase transitions. In dimension d=3, the large deviation is due to a ``distant attracting galaxy'' but a phase transition occurs when f_3(gamma)=1 (at that point, the fluctuations due to individual stars dominate). When d>=5, the large deviation is due to a thin tube (a ``wormhole'') along which the star density increases monotonically, until the point f_d(gamma)=1 (where again fluctuations due to individual stars dominate). In dimension 4 we find a double phase transition, where the transition between low-dimensional behavior (attracting galaxy) and high-dimensional behavior (wormhole) occurs at gamma=4/3. As consequences, we determine the tail behavior of the distance from a star to a uniform point in its cell, and prove a sharp lower bound for the tail probability of the cell's diameter, matching our earlier upper bound.

math.PR↗

arXiv2007

Enumeration formulas for Young tableaux in a diagonal strip

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.

math.CO↗

arXiv2007

Gravitational allocation to Poisson points

For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound: P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d = (d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are positive constants that depend on d and alpha_d. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X>R).

math.PR↗

arXiv2006

Random Sorting Networks

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->infinity 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.

math.PR↗

arXiv2005

Universal finitary codes with exponential tails

In 1977, Keane and Smorodinsky showed that there exists a finitary homomorphism from any finite-alphabet Bernoulli process to any other finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin proved the existence of a finitary homomorphism with finite expected coding length. In this paper, we construct such a homomorphism in which the coding length has exponential tails. Our construction is source-universal, in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions. We also indicate how our methods can be extended to prove a source-specific version of the result for Markov chains.

math.PR↗

arXiv2004

The dynamics of Pythagorean triples

We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning's ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker, that the dynamical system can be obtained as a factor map of a cross-section of the geodesic flow on a quotient space of the hyperbolic plane by the group $Γ(2)$, a free subgroup of the modular group with two generators.

math.DS↗

arXiv2004

Random walks with $k$-wise independent increments

We construct examples of a random walk with pairwise-independent steps which is almost-surely bounded, and for any $m$ and $k$ a random walk with $k$-wise independent steps which has no stationary distribution modulo $m$.

math.PR↗

arXiv2004

Limit shapes for random square Young tableaux and plane partitions

Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random n-by-n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems studied by Logan-Shepp, Vershik-Kerov, and Cohn-Larsen-Propp. Our solution involves methods from the theory of singular integral equations, and sheds light on the somewhat mysterious derivations in these works. An extension to rectangular diagrams, using the same ideas but involving some nontrivial computations, is also given. We give several applications of the main result. First, we show that the location of a particular entry in the tableau is in the limit governed by a semicircle distribution. Next, we derive a result on the length of the longest increasing subsequence in segments of a minimal Erdos-Szekeres permutation, namely a permutation of the numbers 1,2,...,n^2 whose longest monotone subsequence is of length n (and hence minimal by the Erdos-Szekeres theorem). Finally, we prove a limit shape theorem for the surface defined by a random plane partition of a very large integer over a large square (and more generally rectangular) diagram.

math.PR↗

arXiv2003

Waiting for a bat to fly by (in polynomial time)

We observe returns of a simple random walk on a finite graph to a fixed node, and would like to infer properties of the graph, in particular properties of the spectrum of the transition matrix. This is not possible in general, but at least the eigenvalues can be recovered under fairly general conditions, e.g. when the graph has a node-transitive automorphism group. The main result is that by observing polynomially many returns, it is possible to estimate the spectral gap of such a graph up to a constant factor.

math.PR↗

arXiv2003

Shortest paths in the Tower of Hanoi graph and finite automata

We present efficient algorithms for constructing a shortest path between two states 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 63/38 of the largest discs, 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 466/885 between two random points on the Sierpinski gasket of unit side.

math.CO↗

arXiv2003

Integrals, Partitions, and Cellular Automata

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{π^2}{3ab}$$ 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.

math.PR↗

arXiv2003

Explicit formulas for hook walks on continual Young diagrams

We consider, following the work of S. Kerov, random walks which are continuous-space generalizations of the Hook Walks defined by Greene-Nijenhuis-Wilf, performed under the graph of a continual Young diagram. The limiting point of these walks is a point on the graph of the diagram. We present several explicit formulas giving the probability densities of these limiting points in terms of the shape of the diagram. This partially resolves a conjecture of Kerov concerning an explicit formula for the so-called Markov transform. We also present two inverse formulas, reconstructing the shape of the diagram in terms of the densities of the limiting point of the walks. One of these two formulas can be interepreted as an inverse formula for the Markov transform. As a corollary, some new integration identities are derived.

math.PR↗