International Conference on Artificial Intelligence and Statistics, 2020
A common belief in unbiased active learning is that, in order to capture the most informative ins... more A common belief in unbiased active learning is that, in order to capture the most informative instances, the sampling probabilities should be proportional to the uncertainty of the class labels. We argue that this produces suboptimal predictions and present sampling schemes for unbiased pool-based active learning that minimise the actual prediction error, and demonstrate a better predictive performance than competing methods on a number of benchmark datasets. In contrast, both probabilistic and deterministic uncertainty sampling performed worse than simple random sampling on some of the datasets.
We consider two dynamical variants of Dvoretzky's classical problem of random interval coveri... more We consider two dynamical variants of Dvoretzky's classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length ℓ is updated at rate ℓ^-α where α>0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c<2 and there are times at which the circle is not fully covered if and only if c<3. For the Poisson updating model, we obtain analogous results with c<α and c<α+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.
The overhand shuffle is one of the "real" card shuffling methods in the sense that some... more The overhand shuffle is one of the "real" card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37--49] who showed that the mixing time with respect to variation distance is at least of order n^2 and at most of order n^2 n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274--325] to establish a lower bound of order n^2 n, thereby showing that n^2 n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n^3 n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77--85] for the Rudvalis shuffle.
The random-cluster model is a dependent percolation model that has applications in the study of I... more The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter q≥ 1. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where ^ (q) > ^ (q) for q large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs.
Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an important new technique for... more Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson's technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector's problem corresponding to the move-to-front scheme at hand. For the...
This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of in... more This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Z^d and, more generally, on transitive graphs. For iid percolation on Z^d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
Recently, new methods for model assessment, based on subsampling and posterior approximations, ha... more Recently, new methods for model assessment, based on subsampling and posterior approximations, have been proposed for scaling leave-one-out cross-validation (LOO) to large datasets. Although these methods work well for estimating predictive performance for individual models, they are less powerful in model comparison. We propose an efficient method for estimating differences in predictive performance by combining fast approximate LOO surrogates with exact LOO subsampling using the difference estimator and supply proofs with regards to scaling characteristics. The resulting approach can be orders of magnitude more efficient than previous approaches, as well as being better suited to model comparison.
The random-cluster model is a dependent percolation model that has applications in the study of I... more The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the random-cluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of random-cluster measures for arbitrary graphs, and for unimodular transitive graphs, lack of percolation for the free random-cluster measure at the lower critical value on nonamenable graphs, and a number of inequalities for the critical values. Some of these inequalities lead to considerations of isoperimetric constants in certain hyperbolic graphs, and the first nontrivial explicit calculations of such constants are obtained. Applications to the Potts model include Bernoullicity in the Z d case at all temperatures, and non-robust phase transition in the case of nonamenable regular graphs.
A Markov chain is a random process {Xt}, where, in discrete time, t ∈ Z+ and in continuous time, ... more A Markov chain is a random process {Xt}, where, in discrete time, t ∈ Z+ and in continuous time, t ∈ R+ and the Xt’s take values in some state space S, such that given the process up to time t, the distribution of the future of the process only depends on Xt. When S is the symmetric group, i.e. the set of permutations of a number of elements, we think of the Markov chain as a card shuffling chain. In this course, all Markov chains will live on a finite state space S. By enumerating the states 1, 2, . . . , |S|, we may identify S with [|S|] = {1, 2, . . . , |S|}. A Markov chain in discrete time, {X0, X1, X2, . . .} is governed by the starting distribution P(X0 ∈ ·) and the transition matrix P = [pij]i,j∈S where the pij’s are the transition probabilities
The cover time of a finite connected graph is the expected number of steps needed for a simple ra... more The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (
Let $F$ be a probability distribution with support on the non-negative integers. A model is propo... more Let $F$ be a probability distribution with support on the non-negative integers. A model is proposed for generating stationary simple graphs on $\mathbb{Z}$ with degree distribution $F$ and it is shown for this model that the expected total length of all edges at a given vertex is finite if $F$ has finite second moment. It is not hard to see that any stationary model for generating simple graphs on $\mathbb{Z}$ will give infinite mean for the total edge length per vertex if $F$ does not have finite second moment. Hence, finite second moment of $F$ is a necessary and sufficient condition for the existence of a model with finite mean total edge length.
Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant... more Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant under isometries of $\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in $\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays fast enough, then there are almost surely no lines in $Z$. We also show that if the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$. In the case of the Poisson Boolean model with balls of fixed radius $R$ we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also det...
We consider nearest neighbor weighted random walks on the d-dimensional box [n]d that are governe... more We consider nearest neighbor weighted random walks on the d-dimensional box [n]d that are governed by some function g : [0, 1] → [0,∞), by which we mean that standing at x, a neighbor y of x is picked at random and the walk then moves there with probability (1/2)g(n−1y)/(g(n−1y)+g(n−1x)). We do this for g of the form fmn for some function f which assumed to be analytically well-behaved and where mn → ∞ as n → ∞. This class of walks covers an abundance of interesting special cases, e.g., the mean-field Potts model, posterior collapsed Gibbs sampling for Latent Dirichlet allocation and certain Bayesian posteriors for models in nuclear physics. The following are among the results of this paper: • If f is unimodal with negative definite Hessian at its global maximum, then the mixing time of the random walk is O(n log n). • If f is multimodal, then the mixing time is exponential in n, but we show that there is a simulated annealing scheme governed by fK for an increasing sequence of K th...
We study the volatility of the output of a Boolean function when the input bits undergo a natural... more We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For n = 1, 2, . . ., let f n : {0, 1} mn → {0, 1} be a Boolean function and X (n) (t) = (X 1 (t), . . . , X mn (t)) t∈[0,∞) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jump from 0 to 1 with rate p n ∈ [0, 1] and from 1 to 0 with rate q n = 1p n . Our object of study will be C n which is the number of state changes of f n (X (n) (t)) as a function of t during [0, 1]. We say that the family {f n } n≥1 is volatile if C n → ∞ in distribution as n → ∞ and say that {f n } n≥1 is tame if {C n } n≥1 is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that P(C n = 0) → 1 as n → ∞. Finally, we investigate these properties for the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees in various regimes.
Consider a two-person general-sum game on n× n payoff random matrices A and B with iid continuous... more Consider a two-person general-sum game on n× n payoff random matrices A and B with iid continuous entries, for large n. It is shown that the probability that there exists a pure strategy Nash equilibrium that is not pure Pareto optimal remains bounded away from 0 ...
Consider the interchange process on a connected graph G = (V, E) on n vertices. I.e. shuffle a de... more Consider the interchange process on a connected graph G = (V, E) on n vertices. I.e. shuffle a deck of cards by first placing one card at each vertex of G in a fixed order and then at each tick of the clock, picking an edge uniformly at random and switching the two cards at the end vertices of the edge with probability 1/2. Well known special cases are the random transpositions shuffle, where G is the complete graph, and the transposing neighbors shuffle, where G is the n-path. Other cases that have been studied are the d-dimensional grid, the hypercube, lollipop graphs and Erdős-Rényi random graphs above the threshold for connectedness. In this paper the problem is studied for general G. Special attention is focused on trees, random trees and the giant component of critical and supercritical G(N, p) random graphs. Upper and lower bounds on the mixing time are given. In many of the cases, we establish the exact order of the mixing time. We also mention the cases when G is the hypercube and when G is a bounded-degree expander, giving upper and lower bounds on the mixing time.
The overhand shuffle is one of the "real" card shuffling methods in the sense that some people ac... more The overhand shuffle is one of the "real" card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37-49] who showed that the mixing time with respect to variation distance is at least of order n 2 and at most of order n 2 log n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274-325] to establish a lower bound of order n 2 log n, thereby showing that n 2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the (n 3 log n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77-85] for the Rudvalis shuffle.
The overhand shuffle is one of the “real ” card shuffling methods in the sense that some people a... more The overhand shuffle is one of the “real ” card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [4] who showed that the mixing time with respect to variation distance is at least of order n 2 and at most of order n 2 log n. In this paper we use an extension of a lemma of Wilson [6] to establish a lower bound of order n 2 log n, thereby showing that n 2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson’s Lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n 3 log n) lower bound of Wilson [5] for the Rudvalis shuffle. 1
International Conference on Artificial Intelligence and Statistics, 2020
A common belief in unbiased active learning is that, in order to capture the most informative ins... more A common belief in unbiased active learning is that, in order to capture the most informative instances, the sampling probabilities should be proportional to the uncertainty of the class labels. We argue that this produces suboptimal predictions and present sampling schemes for unbiased pool-based active learning that minimise the actual prediction error, and demonstrate a better predictive performance than competing methods on a number of benchmark datasets. In contrast, both probabilistic and deterministic uncertainty sampling performed worse than simple random sampling on some of the datasets.
We consider two dynamical variants of Dvoretzky's classical problem of random interval coveri... more We consider two dynamical variants of Dvoretzky's classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length ℓ is updated at rate ℓ^-α where α>0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c<2 and there are times at which the circle is not fully covered if and only if c<3. For the Poisson updating model, we obtain analogous results with c<α and c<α+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.
The overhand shuffle is one of the "real" card shuffling methods in the sense that some... more The overhand shuffle is one of the "real" card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37--49] who showed that the mixing time with respect to variation distance is at least of order n^2 and at most of order n^2 n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274--325] to establish a lower bound of order n^2 n, thereby showing that n^2 n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n^3 n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77--85] for the Rudvalis shuffle.
The random-cluster model is a dependent percolation model that has applications in the study of I... more The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter q≥ 1. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where ^ (q) > ^ (q) for q large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs.
Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an important new technique for... more Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson's technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector's problem corresponding to the move-to-front scheme at hand. For the...
This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of in... more This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Z^d and, more generally, on transitive graphs. For iid percolation on Z^d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
Recently, new methods for model assessment, based on subsampling and posterior approximations, ha... more Recently, new methods for model assessment, based on subsampling and posterior approximations, have been proposed for scaling leave-one-out cross-validation (LOO) to large datasets. Although these methods work well for estimating predictive performance for individual models, they are less powerful in model comparison. We propose an efficient method for estimating differences in predictive performance by combining fast approximate LOO surrogates with exact LOO subsampling using the difference estimator and supply proofs with regards to scaling characteristics. The resulting approach can be orders of magnitude more efficient than previous approaches, as well as being better suited to model comparison.
The random-cluster model is a dependent percolation model that has applications in the study of I... more The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the random-cluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of random-cluster measures for arbitrary graphs, and for unimodular transitive graphs, lack of percolation for the free random-cluster measure at the lower critical value on nonamenable graphs, and a number of inequalities for the critical values. Some of these inequalities lead to considerations of isoperimetric constants in certain hyperbolic graphs, and the first nontrivial explicit calculations of such constants are obtained. Applications to the Potts model include Bernoullicity in the Z d case at all temperatures, and non-robust phase transition in the case of nonamenable regular graphs.
A Markov chain is a random process {Xt}, where, in discrete time, t ∈ Z+ and in continuous time, ... more A Markov chain is a random process {Xt}, where, in discrete time, t ∈ Z+ and in continuous time, t ∈ R+ and the Xt’s take values in some state space S, such that given the process up to time t, the distribution of the future of the process only depends on Xt. When S is the symmetric group, i.e. the set of permutations of a number of elements, we think of the Markov chain as a card shuffling chain. In this course, all Markov chains will live on a finite state space S. By enumerating the states 1, 2, . . . , |S|, we may identify S with [|S|] = {1, 2, . . . , |S|}. A Markov chain in discrete time, {X0, X1, X2, . . .} is governed by the starting distribution P(X0 ∈ ·) and the transition matrix P = [pij]i,j∈S where the pij’s are the transition probabilities
The cover time of a finite connected graph is the expected number of steps needed for a simple ra... more The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (
Let $F$ be a probability distribution with support on the non-negative integers. A model is propo... more Let $F$ be a probability distribution with support on the non-negative integers. A model is proposed for generating stationary simple graphs on $\mathbb{Z}$ with degree distribution $F$ and it is shown for this model that the expected total length of all edges at a given vertex is finite if $F$ has finite second moment. It is not hard to see that any stationary model for generating simple graphs on $\mathbb{Z}$ will give infinite mean for the total edge length per vertex if $F$ does not have finite second moment. Hence, finite second moment of $F$ is a necessary and sufficient condition for the existence of a model with finite mean total edge length.
Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant... more Suppose that $Z$ is a random closed subset of the hyperbolic plane $\H^2$, whose law is invariant under isometries of $\H^2$. We prove that if the probability that $Z$ contains a fixed ball of radius 1 is larger than some universal constant $p<1$, then there is positive probability that $Z$ contains (bi-infinite) lines. We then consider a family of random sets in $\H^2$ that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let $f(r)$ be the probability that a line segment of length $r$ is contained in such a set $Z$. We show that if $f(r)$ decays fast enough, then there are almost surely no lines in $Z$. We also show that if the decay of $f(r)$ is not too fast, then there are almost surely lines in $Z$. In the case of the Poisson Boolean model with balls of fixed radius $R$ we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also det...
We consider nearest neighbor weighted random walks on the d-dimensional box [n]d that are governe... more We consider nearest neighbor weighted random walks on the d-dimensional box [n]d that are governed by some function g : [0, 1] → [0,∞), by which we mean that standing at x, a neighbor y of x is picked at random and the walk then moves there with probability (1/2)g(n−1y)/(g(n−1y)+g(n−1x)). We do this for g of the form fmn for some function f which assumed to be analytically well-behaved and where mn → ∞ as n → ∞. This class of walks covers an abundance of interesting special cases, e.g., the mean-field Potts model, posterior collapsed Gibbs sampling for Latent Dirichlet allocation and certain Bayesian posteriors for models in nuclear physics. The following are among the results of this paper: • If f is unimodal with negative definite Hessian at its global maximum, then the mixing time of the random walk is O(n log n). • If f is multimodal, then the mixing time is exponential in n, but we show that there is a simulated annealing scheme governed by fK for an increasing sequence of K th...
We study the volatility of the output of a Boolean function when the input bits undergo a natural... more We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For n = 1, 2, . . ., let f n : {0, 1} mn → {0, 1} be a Boolean function and X (n) (t) = (X 1 (t), . . . , X mn (t)) t∈[0,∞) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jump from 0 to 1 with rate p n ∈ [0, 1] and from 1 to 0 with rate q n = 1p n . Our object of study will be C n which is the number of state changes of f n (X (n) (t)) as a function of t during [0, 1]. We say that the family {f n } n≥1 is volatile if C n → ∞ in distribution as n → ∞ and say that {f n } n≥1 is tame if {C n } n≥1 is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that P(C n = 0) → 1 as n → ∞. Finally, we investigate these properties for the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees in various regimes.
Consider a two-person general-sum game on n× n payoff random matrices A and B with iid continuous... more Consider a two-person general-sum game on n× n payoff random matrices A and B with iid continuous entries, for large n. It is shown that the probability that there exists a pure strategy Nash equilibrium that is not pure Pareto optimal remains bounded away from 0 ...
Consider the interchange process on a connected graph G = (V, E) on n vertices. I.e. shuffle a de... more Consider the interchange process on a connected graph G = (V, E) on n vertices. I.e. shuffle a deck of cards by first placing one card at each vertex of G in a fixed order and then at each tick of the clock, picking an edge uniformly at random and switching the two cards at the end vertices of the edge with probability 1/2. Well known special cases are the random transpositions shuffle, where G is the complete graph, and the transposing neighbors shuffle, where G is the n-path. Other cases that have been studied are the d-dimensional grid, the hypercube, lollipop graphs and Erdős-Rényi random graphs above the threshold for connectedness. In this paper the problem is studied for general G. Special attention is focused on trees, random trees and the giant component of critical and supercritical G(N, p) random graphs. Upper and lower bounds on the mixing time are given. In many of the cases, we establish the exact order of the mixing time. We also mention the cases when G is the hypercube and when G is a bounded-degree expander, giving upper and lower bounds on the mixing time.
The overhand shuffle is one of the "real" card shuffling methods in the sense that some people ac... more The overhand shuffle is one of the "real" card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37-49] who showed that the mixing time with respect to variation distance is at least of order n 2 and at most of order n 2 log n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274-325] to establish a lower bound of order n 2 log n, thereby showing that n 2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the (n 3 log n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77-85] for the Rudvalis shuffle.
The overhand shuffle is one of the “real ” card shuffling methods in the sense that some people a... more The overhand shuffle is one of the “real ” card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [4] who showed that the mixing time with respect to variation distance is at least of order n 2 and at most of order n 2 log n. In this paper we use an extension of a lemma of Wilson [6] to establish a lower bound of order n 2 log n, thereby showing that n 2 log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson’s Lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n 3 log n) lower bound of Wilson [5] for the Rudvalis shuffle. 1
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Papers by Johan Jonasson