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November 2024
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4 Reads
P\'olya trees are rooted, unlabeled trees on n vertices. This paper gives an efficient, new way to generate P\'olya trees. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with `reality'. Along the way, we give a product formula for the number of rooted labeled trees preserved by a given automorphism; this refines Cayley's formula.
August 2024
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5 Reads
Let be a group of permutations of kn objects which permutes things independently in disjoint blocks of size k and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of . This includes novel limit theorems for fixed points, cycles of various lengths, number of cycles and inversions. The limits are compound Poisson distributions with interesting dependence structure.
August 2024
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1 Read
Let be the group of permutations of that permutes the first k symbols arbitrarily, then the next k symbols and so on through the last k symbols. Finally the n blocks of size k are permuted in an arbitrary way. For chosen uniformly in , let be the length of the longest increasing subsequence in . For k,n growing, we determine that the limiting mean of is asymptotic to . This is different from parallel variations of the Vershik-Kerov theorem for colored permutations.
August 2023
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6 Reads
Journal of Algebra
June 2023
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16 Reads
The Tsetlin library is a well-studied Markov chain on the symmetric group . It has stationary distribution the Luce model, a nonuniform distribution on , which appears in psychology, horse race betting, and tournament poker. Simple enumerative questions, such as ``what is the distribution of the top k cards?'' or ``what is the distribution of the bottom k cards?'' are long open. We settle these questions and draw attention to a host of parallel questions on the extension to the chambers of a hyperplane arrangement.
May 2023
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12 Reads
For , let be the law of the number of fixed points of a random permutation of . Let be a Poisson law of parameter 1.A classical result shows that converges to for large N and indeed in total variation This implies that and can be coupled to at least this accuracy. This paper constructs such a coupling (a long open problem) using the machinery of intertwining of two Markov chains. This method shows promise for related problems of random matrix theory.
May 2023
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3 Reads
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2 Citations
Journal of Combinatorial Theory Series B
April 2023
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13 Reads
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3 Citations
The Annals of Applied Probability
March 2023
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9 Reads
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4 Citations
January 2023
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50 Reads
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4 Citations
Forum of Mathematics Sigma
Let G be a finite group. Let be subgroups of G and the double coset space. If Q is a probability on G which is constant on conjugacy classes ( ), then the random walk driven by Q on G projects to a Markov chain on . This allows analysis of the lumped chain using the representation theory of G . Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on onto a Markov chain on via the Bruhat decomposition. The chain on has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains with G a compact group are discussed.
... Other techniques to study mixing times were developed in the following years, notably by Aldous and Diaconis [Ald83,AD86], and mixing properties of emblematic card shufflings were precisely understood, also for small decks of cards [BD92]. We refer to [LP17] for an introduction to mixing times and to [DF23] for the mathematics of card shuffling. ...
March 2023
... As noted by Chatterjee and Diaconis [6], this poses a conundrum. On one hand, if X, Y are independent G(∞, 1/2) random graphs then X ∼ = Y a.s. ...
May 2023
Journal of Combinatorial Theory Series B
... If one is ready to abandon accuracy guarantees, also other Monte Carlo estimators become available. In particular, importance sampling (IS) based estimators have appeared to often perform well in practice (Smith and Dawkins 2001;Alimohammadi et al. 2021). Even if the number of samples required for accuracy guarantees is generally too high to be useful, recent advances in sample complexity analysis (Chatterjee and Diaconis 2018) have enabled discovering classes of matrices for which the complexity is provably feasible (Alimohammadi et al. 2021). ...
April 2023
The Annals of Applied Probability
... This problem (in the context of poker tournaments) motivated Diaconis and coauthors [8,9] to study a multi-player version of the gambler's ruin problem that we present here for just 3 players as follows. Starting with initial $ amounts x, y, z ∈ N, at each step of the process two individuals are chosen uniformly at random to make a $1 bet on the outcome of a fair coin toss. ...
Reference:
All In: Give me your money!
August 2022
Statistical Science
... Whenever there is a Markov chain on some state space, one can consider the lumped process on any partitioning of the space. In [11] and [44] the question is studied for double cosets: Given a Markov chain on G, is the lumped process on H \G/K also a Markov chain? An affirmative answer is proven when the Markov chain on G is induced by a probability measure on G which is constant on conjugacy classes. ...
January 2023
Forum of Mathematics Sigma
... First constructed by Ackermann in [1], it was a matter of interest for Erdős and Rényi in [7], Rado in [9], and still attracts many mathematicians, see e.g. [4,5,6,8]. The crucial property needed to define the Random Graph is the following. ...
August 2022
... For relevant background on Markov bases and the connection between statistics and nonlinear algebra, we refer the interested reader to one of the algebraic statistics textbooks Sullivant [2021], Drton et al. [2009]. A notable related body of work is summarized in Diaconis [2022b], where Markov bases are key to formulating partial exchangeability for contingency tables; see also Diaconis [2022a]. ...
August 2022
Scandinavian Journal of Statistics
... It is known [4] that the optimal strategy is not the greedy one as soon as m > 1 and n > 2. Some limiting results were recently obtained in [5] and [13], for instance that the expected optimal score for n m 1 is of the form m + ( √ m) uniformly in n. Results on the fluctuations are currently unknown -except in the case m = 1 -where the limiting distribution has a non-normal behavior as shown in [4]. Since the optimal strategy is rather hard to implement, a fact ultimately due to its connection with permanents [2,6], there has also been some interest in near-optimal strategies that are easier to implement [7]. ...
May 2022
... This was extended in several directions by the first author and Nestoridi [NO22b,NO22a]. Representation-theoretic and orthogonal-polynomial techniques have been used for many card shuffles [Tey20,NO22b,FTW22,NO22a,Nes24], the Ehrenfest urn and a Gibbs sampler [NO22b] as well as multi-allelic Moran mutation-reproduction models [Cor23] Classically probabilistic approaches are used for random walks on Ramanujan and random Cayley graphs [LP16,HO21] as well as repeated averages [Cha+22], and the exclusion process on the circle [Lac16]. Very recently, tools from integrable probability have been used to analyse the asymmetric exclusion processes on the segment [BN22,HS23] and Metropolis biased card shuffling [Zha24]. ...
January 2022
The Annals of Probability
... The author of [43] develops a parameter-free version of classical models for contingency tables, along the lines of de Finetti's notions of partial exchangeability. ...
January 2022
Mathematics
