arXiv subjects

Elchanan Mossel

Publications and source records attributed to Elchanan Mossel.

224 records · Page 13Linked to original sources

arXiv2004

Shuffling by semi-random transpositions

In the cyclic-to-random shuffle, we are given n cards arranged in a circle. At step k, we exchange the k'th card along the circle with a uniformly chosen random card. The problem of determining the mixing time of the cyclic-to-random shuffle was raised by Aldous and Diaconis in 1986. Recently, Mironov used this shuffle as a model for the cryptographic system known as ``RC4'' and proved an upper bound of O(n log n) for the mixing time. We prove a matching lower bound, thus establishing that the mixing time is indeed of order $Θ(n \log n)$. We also prove an upper bound of O(n log n) for the mixing time of any ``semi-random transposition shuffle'', i.e., any shuffle in which a random card is exchanged with another card chosen according to an arbitrary (deterministic or random) rule. To prove our lower bound, we exhibit an explicit complex-valued test function which typically takes very different values for permutations arising from the cyclic-to-random-shuffle and for uniform random permutations; we expect that this test function may be useful in future analysis of RC4. Perhaps surprisingly, the proof hinges on the fact that the function exp(z)-1 has nonzero fixed points in the complex plane. A key insight from our work is the importance of complex analysis tools for uncovering structure in nonreversible Markov chains.

math.PR↗

arXiv2004

Distorted metrics on trees and phylogenetic forests

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these determine the tree and the $d$ length of all edges. Here we consider distortions $\d$ of $d$ such that for all leaves $u$ and $v$ it holds that $|d(u,v) - \d(u,v)| < f/2$ if either $d(u,v) < M$ or $\d(u,v) < M$, where $d$ satisfies $f \leq d(e) \leq g$ for all edges $e$. Given such distortions we show how to reconstruct in polynomial time a forest $T_1,...,T_α$ such that the true tree $T$ may be obtained from that forest by adding $α-1$ edges and $α-1 \leq 2^{-Ω(M/g)} n$. Metric distortions arise naturally in phylogeny, where $d(u,v)$ is defined by the log-det of a covariance matrix associated with $u$ and $v$. of a covariance matrix associated with $u$ and $v$. When $u$ and $v$ are ``far'', the entries of the covariance matrix are small and therefore $\d(u,v)$, which is defined by log-det of an associated empirical-correlation matrix may be a bad estimate of $d(u,v)$ even if the correlation matrix is ``close'' to the covariance matrix. Our metric results are used in order to show how to reconstruct phylogenetic forests with small number of trees from sequences of length logarithmic in the size of the tree. Our method also yields an independent proof that phylogenetic trees can be reconstructed in polynomial time from sequences of polynomial length under the standard assumptions in phylogeny. Both the metric result and its applications to phylogeny are almost tight.

math.CO↗

arXiv2004

Phase transitions in Phylogeny

We apply the theory of markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least 3, and the net transition on each edge is bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2 e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree, having n leaves, can be reconstructed from O(log n) samples (characters) at the leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then there exist topologies which require at least poly(n) samples for reconstruction. Our results are the first rigorous results to establish the role of phase transitions for markov random fields on trees as studied in probability, statistical physics and information theory to the study of phylogenies in mathematical biology.

math.PR↗

arXiv2004

Glauber Dynamics on Trees and Hyperbolic Graphs

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with $n$ vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap $|λ_1-λ_2|$) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time $τ_2$ satisfies $τ_2=O(1)$, then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.

math.PR↗

arXiv2003

New coins from old: computing with unknown bias

Suppose that we are given a function f : (0,1) -> (0,1) and, for some unknown p in (0,1), a sequence of independent tosses of a p-coin (i.e., a coin with probability p of ``heads''). For which functions f is it possible to simulate an f(p)-coin?; This question was raised by S. Asmussen and J. Propp. A simple simulation scheme for the constant function 1/2 was described by von Neumann (1951); this scheme can be easily implemented using a finite automaton. We prove that in general, an f(p)-coin can be simulated by a finite automaton for all p in (0,1), if and only if f is a rational function over Q. We also show that if an f(p)-coin can be simulated by a pushdown automaton, then f is an algebraic function over Q; however, pushdown automata can simulate f(p)-coins for certain non-rational functions such as the square root of p. These results complement the work of Keane and O'Brien (1994), who determined the functions $f$ for which an f(p)-coin can be simulated when there are no computational restrictions on the simulation scheme.

math.PR↗

arXiv2002

Mixing times of the biased card shuffling and the asymmetric exclusion process

Consider the following method of card shuffling. Start with a deck of $N$ cards numbered 1 through N. Fix a parameter $p$ between 0 and 1. In this model a ``shuffle'' consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads then we arrange the two cards so that the lower numbered card comes before the higher numbered card. If the coin comes up tails then we arrange the cards with the higher numbered card first. In this paper we prove that for all p not equal to 1/2, the mixing time of this card shuffling is O(N^2), as conjectured by Diaconis and Ram [DR]. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

math.PR↗

arXiv2002

On the mixing time of simple random walk on the super critical percolation cluster

We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in $\Z^d$. We show that for $d \geq 2$ and $p > p_c(\Z^d)$, the mixing time of simple random walk on the largest cluster inside $\{-n,...,n\}^d$ is $Θ(n^2)$ - thus the mixing time is robust up to constant factor.

math.PR↗

arXiv2001

Information flow on trees

Consider a tree network $T$, where each edge acts as an independent copy of a given channel $M$, and information is propagated from the root. For which $T$ and $M$ does the configuration obtained at level $n$ of $T$ typically contain significant information on the root variable? This problem arose independently in biology, information theory and statistical physics. For all $b$, we construct a channel for which the variable at the root of the $b$-ary tree is independent of the configuration at level 2 of that tree, yet for sufficiently large $B>b$, the mutual information between the configuration at level $n$ of the $B$-ary tree and the root variable is bounded away from zero. This is related to certain secret-sharing protocols. We improve the upper bounds on information flow for asymmetric binary channels (which correspond to the Ising model with an external field) and for symmetric $q$-ary channels (which correspond to Potts models). Let $\lam_2(M)$ denote the second largest eigenvalue of $M$, in absolute value. A CLT of Kesten and Stigum~(1966) implies that if $b |\lam_2(M)|^2 >1$, then the {\em census} of the variables at any level of the $b$-ary tree, contains significant information on the root variable. We establish a converse: if $b |\lam_2(M)|^2 < 1$, then the census of the variables at level $n$ of the $b$-ary tree is asymptotically independent of the root variable. This contrasts with examples where $b |\lam_2(M)|^2 <1$, yet the {\em configuration} at level $n$ is not asymptotically independent of the root variable.

math.PR↗