arXiv subjects

Peter Winkler

Publications and source records attributed to Peter Winkler.

At least 19 recordsLinked to original sources

arXiv2026

Sets that Support a Joint Distribution

Given probability distributions $\mu$ and $\nu$ on measure spaces $X$ and $Y$, and a closed set $S \subseteq X \times Y$, when is there a probability distribution on $X \times Y$ whose marginals are $\mu$ and $\nu$, and whose support is precisely $S$? We answer the question when the marginals are discrete, and when the marginals are continuous distributions on the real line. Of special interest is the case where $S \subseteq [0,1]^2$ and $\mu$ and $\nu$ are Lebesgue measure; then the above question is tantamount to ``when is $S$ the support of a doubly stochastic measure?". The discrete case is generalized to determine when a (possibly infinite) edge-capacitated, node-weighted graph supports a full, nowhere-zero flow; for the continuous case we provide a particularly straightforward characterization when the set in question is regular (i.e., is the closure of its interior).

math.PR↗

arXiv2026

Is There An Ideal Color Wheel?

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors. One might wonder: is there a color wheel in which all six of the sections have this property, without all the sections being the same color? We show that the answer is no, not just for the 6-cycle but for any finite connected graph; indeed, for any finite, strongly connected, edge-weighted digraph. The result generalizes the ``harmonic lemma" for graphs, replacing the well-behaved averaging function by paint blending, about which almost nothing is assumed. Our proof makes use of a Markov chain stopping rule.

math.CO↗

arXiv2024

Block coupling and rapidly mixing k-heights

A $k$-height on a graph $G=(V, E)$ is an assignment $V\to\{0, \ldots, k\}$ such that the value on ajacent vertices differs by at most $1$. We study the Markov chain on $k$-heights that in each step selects a vertex at random, and, if admissible, increases or decreases the value at this vertex by one. In the cases of $2$-heights and $3$-heights we show that this Markov chain is rapidly mixing on certain families of grid-like graphs and on planar cubic $3$-connected graphs. The result is based on a novel technique called block coupling, which is derived from the well-established monotone coupling approach. This technique may also be effective when analyzing other Markov chains that operate on configurations of spin systems that form a distributive lattice. It is therefore of independent interest.

cs.DM↗

arXiv2023

Leading All The Way

Xavier and Yushi run a ``random race'' as follows. A continuous probability distribution $μ$ on the real line is chosen. The runners begin at zero. At time $i$ Xavier draws $\mathbf{X}_i$ from $μ$ and advances that distance, while Yushi advances by an independent drawing $\mathbf{Y}_i$. After $n$ such moves, Xavier wins a valuable prize provided he not only wins the race but leads after every step; that is, $\sum_{i=1}^k \mathbf{X}_i > \sum_{i=1}^k \mathbf{Y}_i$ for all $k = 1,2, \dots, n$. What distribution is best for Xavier, and what then is his probability of getting the prize?

math.PR↗

arXiv2023

Large deviation principle for random permutations

We derive a large deviation principle for random permutations induced by probability measures of the unit square, called permutons. These permutations are called $μ$-random permutations. We also introduce and study a new general class of models of random permutations, called Gibbs permutation models, which combines and generalizes $μ$-random permutations and the celebrated Mallows model for permutations. Most of our results hold in the general setting of Gibbs permutation models. We apply the tools that we develop to the case of $μ$-random permutations conditioned to have an atypical proportion of patterns. Several results are made more concrete in the specific case of inversions. For instance, we prove the existence of at least one phase transition for a generalized version of the Mallows model where the base measure is non-uniform. This is in contrast with the results of Starr (2009, 2018) on the (standard) Mallows model, where the absence of phase transition, i.e., phase uniqueness, was proven. Our results naturally lead us to investigate a new notion of permutons, called conditionally constant permutons, which generalizes both pattern-avoiding and pattern-packing permutons. We describe some properties of conditionally constant permutons with respect to inversions. The study of conditionally constant permutons for general patterns seems to be a challenging problem.

math.PR↗

arXiv2019

Exploration of another Sol Lewitt puzzle from Barry Cipra

At MOVES 2019, Barry Cipra casually introduced a new "Sol Lewitt" puzzle to fellow conference goers. Several brainstorming sessions ensued with Barry, Peter Winkler , Donna Dietz, and other attendees. This paper is to document the puzzle and some insights so others can enjoy and build on this lovely puzzle. (Look for it in an upcoming book by Peter!)

math.HO↗

arXiv2018

The minimum Manhattan distance and minimum jump of permutations

Let $π$ be a permutation of $\{1,2,\ldots,n\}$. If we identify a permutation with its graph, namely the set of $n$ dots at positions $(i,π(i))$, it is natural to consider the minimum $L^1$ (Manhattan) distance, $d(π)$, between any pair of dots. The paper computes the expected value (and higher moments) of $d(π)$ when $n\rightarrow\infty$ and $π$ is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when $d$ is fixed and $n\rightarrow\infty$, the probability that $d(π)\geq d+2$ tends to $e^{-d^2 - d}$. The minimum jump $mj(π)$ of $π$, defined by $mj(π)=\min_{1\leq i\leq n-1} |π(i+1)-π(i)|$, is another natural measure in this context. The paper computes the asymptotic moments of $mj(π)$, and the asymptotic probability that $mj(π)\geq d+1$ for any constant $d$.

math.CO↗

arXiv2018

Abelian logic gates

An abelian processor is an automaton whose output is independent of the order of its inputs. Bond and Levine have proved that a network of abelian processors performs the same computation regardless of processing order (subject only to a halting condition). We prove that any finite abelian processor can be emulated by a network of certain very simple abelian processors, which we call gates. The most fundamental gate is a "toppler", which absorbs input particles until their number exceeds some given threshold, at which point it topples, emitting one particle and returning to its initial state. With the exception of an adder gate, which simply combines two streams of particles, each of our gates has only one input wire. Our results can be reformulated in terms of the functions computed by processors, and one consequence is that any increasing function from N^k to N^l that is the sum of a linear function and a periodic function can be expressed in terms of (possibly nested) sums of floors of quotients by integers.

cs.DM↗

arXiv2018

Mixing Time for Some Adjacent Transposition Markov Chains

We prove rapid mixing for certain Markov chains on the set $S_n$ of permutations on $1,2,\dots,n$ in which adjacent transpositions are made with probabilities that depend on the items being transposed. Typically, when in state $σ$, a position $i<n$ is chosen uniformly at random, and $σ(i)$ and $σ(i{+}1)$ are swapped with probability depending on $σ(i)$ and $σ(i{+}1)$. The stationary distributions of such chains appear in various fields of theoretical computer science, and rapid mixing established in the uniform case. Recently, there has been progress in cases with biased stationary distributions, but there are wide classes of such chains whose mixing time is unknown. One case of particular interest is what we call the "gladiator chain," in which each number $g$ is assigned a "strength" $s_g$ and when $g$ and $g'$ are adjacent and chosen for possible swapping, $g$ comes out on top with probability $s_g/(s_g + s_{g'})$. We obtain a polynomial-time upper bound on mixing time when the gladiators fall into only three strength classes. A preliminary version of this paper appeared as "Mixing of Permutations by Biased Transposition" in STACS 2017.

cs.DS↗

arXiv2015

Permutations with fixed pattern densities

We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed \hbox{12} density, with fixed \hbox{12} and \hbox{123} densities, with fixed \hbox{12} density and the sum of \hbox{123} and \hbox{213} densities, and with fixed \hbox{123} and \hbox{321} densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.

math.CO↗

arXiv2014

Hunter & Mole

We consider a variation of a cops and robbers game in which the cop---here referred to as "hunter"---is not constrained by the graph but must play in the dark against a "mole." We characterize the graphs---which we will call "hunter-win"---on which the hunter can guarantee capture of the mole in bounded time. We also define an optimal hunter strategy (and consequently an upper bound on maximum game time on hunter-win graphs) and note that an optimal hunter strategy need not take advantage of the hunter's unconstrained movement! This game comes from a puzzle of unknown origin which was told to the authors by Dick Hess.

math.CO↗

arXiv2013

Cops vs. Gambler

We consider a variation of cop vs.\ robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on $V(G)$ and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected $n$-vertex graph is exactly $n$. We also give bounds on the (generally greater) expected capture time when the gambler's distribution is unknown to the cop.

math.CO↗

arXiv2013

Avoidance Coupling

We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.

math.PR↗

arXiv2013

Capturing the Drunk Robber on a Graph

We show that the expected time for a smart "cop" to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on $G$; the cop sees all and moves as she wishes, with the object of "capturing" the robber---that is, occupying the same vertex---in least expected time. We show that the cop succeeds in expected time no more than $n + {\rm o}(n)$. Since there are graphs in which capture time is at least $n - o(n)$, this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.

math.CO↗

arXiv2012

Mixing times and moving targets

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states $x$ and moving large sets $(A_s)_s$, of the hitting time of $(A_s)_s$ starting from $x$. We prove that in the case of the $d$-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a "cat and mouse" game.

math.PR↗

arXiv2012

Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with minimal area, (up to constant), namely $Θ(1/\log n)$. As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions $\{B(s): s \ge 0\}$ and $\{W(s): s \ge 0\}$. Let $τ_t:=\min\{s \ge 0: B(s)=t\}$. Then $X_t=W(τ_t)$ is a Cauchy process, and $K:=\{(a,X_t+at) : a,t \in [0,1]\}$ is a Kakeya set of zero area. The area of the $ε$-neighborhood of $K$ is as small as possible, i.e., almost surely of order $Θ(1/|\log ε|)$.

math.PR↗

arXiv2012

The Phase Transition for Dyadic Tilings

A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n->infinity, as conjectured by Joel Spencer in 1999. In particular we prove that if p=7/8, such a tiling exists with probability at least 1-(3/4)^n. The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.

math.PR↗

arXiv2011

Can extra updates delay mixing?

We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations.

math.PR↗