Hopf algebras and Markov chains: two examples and a theory

Diaconis, Persi; Pang, C. Y. Amy; Ram, Arun

doi:10.1007/s10801-013-0456-7

Cited by 26 publications

(39 citation statements)

References 88 publications

(88 reference statements)

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“…Let n P N be given andS ≔ Ppnq, the set of all partitions of n. Thus if n " 4,S " t4, 31, 22, 211, 1111u. An absorbing Markov chain onS, modeled on a rock breaking Markov chain studied by Kolmogorov, is developed in Diaconis, Pang and Ram [9]. Briefly, if λ " pλ 1 , λ 2 , ..., λ l q, with λ 1 ě λ 2 ě¨¨¨ě λ l ą 0, λ 1`¨¨¨`λl " n, the chain proceeds from λ by independently choosing, for i P 1, l , binomial variables λ p1q i of parameters pλ i , 1{2q, so that we can write λ i ≕ λ p1q i`λ p2q i .…”

Section: Example Of Rock Breakingmentioning

confidence: 99%

On quantitative convergence to quasi-stationarity

Diaconis¹,

Miclo²

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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The quantitative long time behavior of absorbing, finite, irreducible Markov processes is considered. Via Doob transforms, it is shown that only the knowledge of the ratio of the values of the underlying first Dirichlet eigenvector is necessary to come back to the well-investigated situation of the convergence to equilibrium of ergodic finite Markov processes. This leads to explicit estimates on the convergence to quasi-stationarity, in particular via functional inequalities. When the process is reversible, the optimal exponential rate consisting of the spectral gap between the two first Dirichlet eigenvalues is recovered. Several simple examples are provided to illustrate the bounds obtained.Keywords: Absorbing finite Markov process, quasi-stationary measure, Doob transform, quantitative bounds on convergence, spectral gap, logarithmic Sobolev inequality, birth and death process.

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Section: Example Of Rock Breakingmentioning

confidence: 99%

On quantitative convergence to quasi-stationarity

Diaconis¹,

Miclo²

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…4], [31] or [37, §9]. Some instances of these operators in the recent literature include [12], [29], and [32]. For references to earlier work on Eulerian idempotents, see [4, §14].…”

Section: Coradical Filtration and Primitive Elementsmentioning

confidence: 99%

“…Similarly, each subspace H (n) is graded with (H (n) ) m = H (n) ∩ H m . Hence, gr H inherits a second grading for which (12) (gr H) m :…”

Section: 5mentioning

confidence: 99%

The characteristic polynomial of the Adams operators on graded connected Hopf algebras

Aguiar

Lauve

2015

Algebra Number Theory

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The Adams operators Ψ n on a Hopf algebra H are the convolution powers of the identity of H. They are also called Hopf powers or Sweedler powers. We study the Adams operators when H is graded connected. The main result is a complete description of the characteristic polynomial-both eigenvalues and their multiplicities-for the action of the operator Ψ n on each homogeneous component of H. The eigenvalues are powers of n. The multiplicities are independent of n, and in fact only depend on the dimension sequence of H. These results apply in particular to the antipode of H, as the case n = −1. We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and for q-Hopf algebras.

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“…Examples are given below and a host of further examples of non-reversible chains with explicit real left and right eigen-vectors are in Diaconis, Pang, and Ram (2011). Of course, without reversibility, the eigen-vectors may take complex values and need not be othogonal, but they satisfy the biorthogonality relationship…”

Section: Non-reversible Chains and Biorthogonal Expansionsmentioning

confidence: 99%