A phase transition in the random transposition random walk
Nathanael Berestycki, Rick Durrett
备注：
Revisions include considerable changes in the presentation of section 6 (proof of the CLT in the supercritical regime), and several typos corrected. Also, the figures are now available as a separate .ps file
Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on $n$ elements. Consider this walk in continuous time starting at the identity and let $D_t$ be the minimum number of transpositions needed to go back to the identity from the location at time $t$. $D_t$ undergoes a phase transition: the distance $D_{cn/2} \sim u(c)n$, where $u$ is an explicit function satisfying $u(c)=c/2$ for $c \le 1$ and $u(c)1$. In other words, the distance to the identity is roughly linear during the subcritical phase, and after critical time $n/2$ it becomes sublinear. In addition, we describe the fluctuations of $D_{cn/2}$ about its mean in each of the threeregimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the \Erdős-Renyi random graph model.
This paper explores the mixing time of the random transposition walk on the symmetric group. While it has long been known that this walk mixes in order n*log(n) time, this result has not previously been attained using coupling. A coupling argument showing the correct order mixing time is presented. This is accomplished by first projecting to conjugacy classes, and then using the Bubley-Dyer path coupling construction. In order to obtain appropriate bounds on the time it takes the path coupling to meet, ideas from Schramm's paper "Compositions of Random Transpositions" are used.
We show that the random transposition walk on the symmetric group $S_n$ has cutoff in separation distance at $\frac{1}{2}n \log n$, by constructing a strong stationary time. The construction involves working with cycle types of permutations and some partition combinatorics.
In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff…
Several cycle lexicographical orders are found to describe the relative likelihood of elements of the random walks on the symmetric group generated by the conjugacy classes of transpositions, 3-cycles, and n-cycles. Spectral analysis finds sufficient time for the orders to hold. This partially answers a conjecture that the n-cycles are the least likely elements of the transposition walk on the symmetric group. A likelihood order contributes to understanding the total variation distance and separation distance for a random walk.
This paper explores the mixing time of the random transposition walk on permutations with one-sided interval restrictions. In particular, we're interested in the notion of cutoff, a phenomenon which occurs when mixing occurs in a window of order smaller than the mixing time. One of the main tools of the paper is the diagonalization obtained by Hanlon; the use of the spectral information is inspired by the famous paper of Diaconis and Shahshahani on the mixing time of the random transposition walk on the entire symmetric group. The diagonalization allows us to prove chi-squared cutoff for a bro…
Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the random walk slows down abruptly (i.e., the acceleration drops from 0 to -\infty at this time as n tends to \infty). On the other hand, the largest cycle size changes from microscopic to giant. The proof of this last result is both considerably simpler and more general than in a previous result of Oded Schramm (2005) for random transpositions. It turns out t…
We prove sharp rates of convergence to the Ewens equilibrium distribution for a family of Metropolis algorithms based on the random transposition shuffle on the symmetric group, with starting point at the identity. The proofs rely heavily on the theory of symmetric Jack polynomials, developed initially by Jack [Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971) 1-18], Macdonald [Symmetric Functions and Hall Polynomials (1995) New York] and Stanley [Adv. Math. 77 (1989) 76-115]. This completes the analysis started by Diaconis and Hanlon in [Contemp. Math. 138 (1992) 99-117]. In the end we also ex…
We study the rate of convergence of the Markov chain on $S_n$ which starts with a random $(n-k)$-cycle for a fixed $k \geq 1$, followed by random transpositions. The convergence to the stationary distribution turns out to be of order $n$. We show that after $cn + \frac{\ln k}{2}n$ steps for $c>0$, the law of the Markov chain is close to the uniform distribution. The character of the defining representation is used as test function to obtain a lower bound for the total variation distance. We identify the asymptotic distribution of the test function given the law of the Markov chain for the $(n-…
This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here, we study three new examples. It is trivial that the distance for random walk on the hypercube is smooth and is given by one simple formula. In the case of random adjacent transpositions, we find that there is no phase transition even though the distance has different scalings in three different regimes. In the case of a random 3-regular graph, there is a ph…
In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $λ_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $λ_n=\frac{1+ε_n}{n-1}$ where $n^{-{1/3}+δ}\le ε_n<1$ and $δ>0$, a random induced subgraph has a.s. a unique largest component of size $\wp(ε_n)\frac{1+ε_n}{n-1}n!$, where $\wp(ε_n)$ is the survival probability of a specific branchi…
We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters.
We study random typical minimal factorizations of the $n$-cycle, which are factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions, chosen uniformly at random. Our main result is, roughly speaking, a local convergence theorem for the trajectories of finitely many points in the factorization. The main tool is an encoding of the factorization by an edge and vertex-labelled tree, which is shown to converge to Kesten's infinite Bienaymé-Galton-Watson tree with Poisson offspring distribution, uniform i.i.d. edge labels and vertex labels obtained by a local exploration algorithm.
Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $σ_t$ be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of the distance from the identity at time $cn/2$ has a phase transition at $c=1$. Here we investigate some consequences of this result for the geometry of $G_n$. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occurs at…
We present an improved version of Diaconis' upper bound lemma, which is used to compute the limiting value of the distance to stationarity. We then apply it to random transpositions studied by Diaconis and Shahshahani.
We study the asymptotic behaviour of random factorizations of the $n$-cycle into transpositions of fixed genus $g>0$. They have a geometric interpretation as branched covers of the sphere and their enumeration as Hurwitz numbers was extensively studied in algebraic combinatorics and enumerative geometry. On the probabilistic side, several models and properties of permutation factorizations were studied in previous works, in particular minimal factorizations of cycles into transpositions (which corresponds to the case $g=0$ of this work). Using the representation of factorizations via unicellul…
A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider $d$ many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree $2d$. We consider the following dynamics. The dimension (i.e. size) of each permutation grows by coupled Chinese Restaurant Processes, while in time each permutation evolves according to the random transposition chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in…
We show that for any semi-random transposition shuffle on $n$ cards, the mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that there is cutoff at this time with a window of size O(n), provided further that $k\to\infty$ as $n\to\infty$ (and no cutoff otherwise). For the random-to-random transposition shuffle we show cutoff at time $(1/2)n\log k$ for the same conditions on $k$. Finally, we analyse the cyclic-to-random transposition shuffle and show partial mixing occurs at time $\leαn\log k$ …
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 mixi…
We study random typical minimal factorizations of the $n$-cycle into transpositions, which are factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions. By viewing transpositions as chords of the unit disk and by reading them one after the other, one obtains a sequence of increasing laminations of the unit disk (i.e. compact subsets of the unit disk made of non-intersecting chords). When an order of $\sqrt{n}$ consecutive transpositions have been read, we establish, roughly speaking, that a phase transition occurs and that the associated laminations converge to a new one-paramete…
Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We define two cyclic to random transposition shuffles. The first one works as follows: for $j=1,..., n$, on the $j$-th step transpose the card that was \it originally\rm\ the $j$-th from the left with a random card (possibly itself). The second shuffle works as follows: on the $j$-th step, transpose the card that is \it currently\rm\ in the $j$-th position from the left with a random card (possibly itself…
Let $Y=(y_1,y_2,...)$, $y_1\ge y_2\ge...$, be the list of sizes of the cycles in the composition of $c n$ transpositions on the set $\{1,2,...,n\}$. We prove that if $c>1/2$ is constant and $n\to\infty$, the distribution of $f(c)Y/n$ converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function $f$ is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process.
The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this paper to mix for $\frac{1}{2} \leq p \leq 1$ fixed, $n$ sufficiently large in between $\log_{1/p}(n)$ steps and $\log_{2/(1+p)}(n)$ steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric…
We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.
We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $c\le 1$ an…
We prove that the limit profile of star transpositions at time $t= n \log n +cn$ is equal to $d_{\text{T.V.}}(\text{Poiss}(1+e^{-c}), \text{Poiss}(1))$. We prove this by developing a technique for comparing the limit profile behavior of two reversible Markov chains on the same space, that share the same stationary distribution and eigenbasis. We then compare the limit profile of star transpositions to the limit profile of random transpositions, as studied in \cite{Teyssier}, and prove that they have the same limit profile at the respective cutoff times.
We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff for the one-sided transpo…
We characterize the minimum-length sequences of independent lazy simple transpositions whose composition is a uniformly random permutation. For every reduced word of the reverse permutation there is exactly one valid way to assign probabilities to the transpositions. It is an open problem to determine the minimum length of such a sequence when the simplicity condition is dropped.
Let W be a Wishart random matrix of size d^2 times d^2, considered as a block matrix with d times d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d tends to infinity. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "W…
Contingency tables are useful objects in statistics for representing 2-way data. With fixed row and column sums, and a total of $n$ entries, contingency tables correspond to parabolic double cosets of $S_n$. The uniform distribution on $S_n$ induces the Fisher-Yates distribution, classical for its use in the chi-squared test for independence. A Markov chain on $S_n$ can then induce a random process on the space of contingency tables through the double cosets correspondence. The random transpositions Markov chain on $S_n$ induces a natural `swap' Markov chain on the space of contingency tables;…
Let $X_{t}$ and $Y_{t}$ be two Markov chains, on state spaces $Ω\subset \hatΩ$. In this paper, we discuss how to prove bounds on the spectrum of $X_{t}$ based on bounds on the spectrum of $Y_{t}$. This generalizes work of Diaconis, Saloff-Coste, Yuen and others on comparison of chains in the case $Ω= \hatΩ$. The main tool is the extension of functions from the smaller space to the larger, which allows comparison of the entire spectrum of the two chains. The theory is used to give quick analyses of several chains without symmetry. The main application is to a `random transposition' walk on dera…
We give refined estimates for the discrete time and continuous time versions of some basic random walks on the symmetric and alternating groups $S_n$ and $A_n$. We consider the following models: random transposition, transpose top with random, random insertion, and walks generated by the uniform measure on a conjugacy class. In the case of random walks on $S_n$ and $A_n$ generated by the uniform measure on a conjugacy class, we show that in continuous time the $\ell^2$-cuttoff has a lower bound of $(n/2)\log n$. This result, along with the results of Müller, Schlage-Puchta and Roichman, demons…
Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositi…
We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the number of cycles in the commutator $[σ,τ] = στσ^{-1} τ^{-1}$ where $σ$ is uniformly distributed, and $τ$ is either one cycle, the product of two cycles of same size, or the product of many transpositions.
Consider the process of random transpositions on the complete graph. We use representation theory to give an exact, simple formula for the expected number of cycles of size k at time t, in terms of an incomplete Beta function. Using this we show that the expected number of cycles of size k jumps from 0 to its equilibrium value, 1/k, at the time where the giant component of the associated random graph first exceeds k. Consequently we deduce a new and simple proof of Schramm's theorem on random transpositions, that giant cycles emerge at the same time as the giant component in the random graph. …
In the present paper we define conservative and semiconservative random walks in $\mathbb{Z}^d$ and study different families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (Pólya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in $\mathbb{Z}^d$ by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in $\mathbb{Z}^d$ for any $d\geq3$ and transient random walks in $\mathbb{…
We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m>x_1>x_2>...>x_n>0, which is a rescaled alcove of the affine Weyl group C_n. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0,m). Another case models n non-collid…
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 $m$.
We show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random fixed-point-free involutions only, a.k.a. random perfect matchings on the complete graph. Our data indicate that the algorithms generate random samples with the expected distribution of cycle lengths, which we derive, and for relatively small samples, which can actually be very large in absolute numbers, we argue that they generate samples indistinguishable …
We consider a process of noncolliding $q$-exchangeable random walks on $\mathbb{Z}$ making steps $0$ (straight) and $-1$ (down). A single random walk is called $q$-exchangeable if under an elementary transposition of the neighboring steps (down, straight) $\to$ (straight, down) the probability of the trajectory is multiplied by a parameter $q\in(0,1)$. Our process of $m$ noncolliding $q$-exchangeable random walks is obtained from the independent $q$-exchangeable walks via the Doob's $h$-transform for a certain nonnegative eigenfunction $h$ with the eigenvalue less than $1$. The system of $m$ w…
We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker visible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger th…
We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, with elements of ${0,1}$ given by i.i.d. fair coin flips on the range $\mathcal {R}$ of a random walk $X$ run until time $T$ and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold $T=1/2T_{\mathrm {cov}}(\mathbf {Z}_n^d)$. As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph $\mathbf {Z}_2\wr \mathbf {Z}_n^d$, $d\geq3$, has a cutoff with threshold $1/2T_{\mathrm {cov}}(\mathbf {Z}_n^d)$. We give a general criteri…
In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime.
We consider a random walk $(Z^{(1)}_n, ..., Z^{(K+1)}_n) \in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor $σ_K^2 = \frac{2}{K+2}$ with respect to the case of the classical simple random walk without constraint.
