On meteors, earthworms and WIMPs

Billey, Sara; Burdzy, Krzysztof; Pal, Soumik; Sagan, Bruce E.

doi:10.1214/14-aap1035

Cited by 15 publications

(24 citation statements)

References 29 publications

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“…For the graph in Figure 2.3 (a), we have: In the last iteration, we simply replace the last unlabeled vertex with 1, so we obtain two labelings in this case, namely [4, 3, 2, 1] and [4,3,1,2]. Similarly, for the graph in For Figure 2.2 (b) we run the same algorithm.…”

Section: Recursive Constructionmentioning

confidence: 99%

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Counting Peaks on Graphs

Everham¹,

Marcantonio²,

Insko³

2019

Aquila

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Given a graph G with n vertices and a bijective labeling of the vertices using the integers 1, 2, . . . , n, we say G has a peak at vertex v if the degree of v is greater than or equal to 2 and if the label on v is larger than the label of all its neighbors. For each subset S ⊂ V(G), we want to determine the number of distinct labelings of the vertices of G such that the vertices in S are precisely the peaks of G. The set S is called the peak set of the graph G, and the set of all labelings with peaks at every vertex in S is denoted by P(S ; G). In this paper, we present an algorithm for constructing all of the labelings in P(S ; G) and explore special cases of peak sets in certain families of graphs, specifically those related to the joins of graphs. We note that this work is a direct generalization of the research of peak sets of permutations, which is the special case when G is the path graph on n vertices.

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Section: Recursive Constructionmentioning

confidence: 99%

“…Their investigations were essentially restricted to the study of peak sets on paths satisfying certain symmetry constraints about the middle vertex. In 2014, the theory developed by Billey, Burdzy, and Sagan was employed to create a probabilistic mass redistribution model on graphs [1].…”

Section: Introductionmentioning

confidence: 99%

Counting Peaks on Graphs

Everham¹,

Marcantonio²,

Insko³

2019

Aquila

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“…Recently, [BBPS15] studied the potlatch process on finite graphs under the name of meteor process and [Bur15] revisited the lattice case. Among other things, [BBPS15] investigated the rate of convergence of the potlatch process on a finite graph in L 1 -Wasserstein distance. For the discrete d-dimensional torus T d n := (Z/nZ) d they obtained an O(n 2 ) bound on the relaxation time (see [BBPS15,Thm.…”

Section: Introductionmentioning

confidence: 99%

“…Among other things, [BBPS15] investigated the rate of convergence of the potlatch process on a finite graph in L 1 -Wasserstein distance. For the discrete d-dimensional torus T d n := (Z/nZ) d they obtained an O(n 2 ) bound on the relaxation time (see [BBPS15,Thm. 3.6]).…”

Section: Introductionmentioning

confidence: 99%

Rates of convergence to equilibrium for Potlatch and Smoothing processes

Banerjee¹,

Burdzy²

2020

Preprint

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We analyze the local and global smoothing rates of the smoothing process and obtain convergence rates to stationarity for the dual process known as the potlatch process. For general finite graphs, we connect the smoothing and convergence rates to the spectral gap of the associated Markov chain. We perform a more detailed analysis of these processes on the torus. Polynomial corrections to the smoothing rates are obtained. They show that local smoothing happens faster than global smoothing. These polynomial rates translate to rates of convergence to stationarity in L 2 -Wasserstein distance for the potlatch process on Z d . 2010 Mathematics Subject Classification. 60K35.

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“…For any nadmissible set S, they found these cardinalities satisfy |P (S; n)| = p(n)2 n−|S|−1 (1) where |S| denotes the cardinality of the set S, and where the peak polynomial p(n) is a polynomial of degree max(S) − 1 that takes integral values when evaluated at integers [10, Theorem 1.1]. Their study was motivated by a problem in probability theory which explored the mass distribution on graphs as it relates to random permutations with specific peak sets; this research was presented in [9]. Billey, Burdzy, and Sagan also computed closed formulas for the peak polynomials p(n) for various special cases of P (S; n) using the method of finite differences, and Billey, Fahrbach, and Talmage then studied the coefficients and zeros of peak polynomials [11].…”

Section: Introductionmentioning

confidence: 99%

Peak sets of classical Coxeter groups

Diaz-Lopez

Harris

Insko

et al. 2017

Involve

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We say a permutation π = π1π2 · · · πn in the symmetric group Sn has a peak at index i if πi−1 < πi > πi+1 and we let P (π) = {i ∈ {1, 2, . . . , n} | i is a peak of π}. Given a set S of positive integers, we let P (S; n) denote the subset of Sn consisting of all permutations π, where P (π) = S. In 2013, Billey, Burdzy, and Sagan proved |P (S; n)| = p(n)2 n−|S|−1 , where p(n) is a polynomial of degree max(S) − 1. In 2014, CastroVelez et al. considered the Coxeter group of type Bn as the group of signed permutations on n letters and showed that |PB(S; n)| = p(n)2 2n−|S|−1 where p(n) is the same polynomial of degree max(S) − 1. In this paper we partition the sets P (S; n) ⊂ Sn studied by Billey, Burdzy, and Sagan into subsets of P (S; n) of permutations with peak set S that end with an ascent to a fixed integer k or a descent and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie type Cn and Dn into S2n, we partition these groups into bundles of permutations π1π2 · · · πn|πn+1 · · · π2n such that π1π2 · · · πn has the same relative order as some permutation σ1σ2 · · · σn ∈ Sn. This allows us to count the number of permutations in types Cn and Dn with a given peak set S by reducing the enumeration to calculations in the symmetric group and sums across the rows of Pascal's triangle.

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On meteors, earthworms and WIMPs

Cited by 15 publications

References 29 publications

Counting Peaks on Graphs

Counting Peaks on Graphs

Rates of convergence to equilibrium for Potlatch and Smoothing processes

Peak sets of classical Coxeter groups

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