609 research outputs found
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Combable functions, quasimorphisms, and the central limit theorem
A function on a discrete group is weakly combable if its discrete derivative
with respect to a combing can be calculated by a finite state automaton. A
weakly combable function is bicombable if it is Lipschitz in both the left and
right invariant word metrics.
Examples of bicombable functions on word-hyperbolic groups include (i)
homomorphisms to Z (ii) word length with respect to a finite generating set
(iii) most known explicit constructions of quasimorphisms (e.g. the
Epstein-Fujiwara counting quasimorphisms)
We show that bicombable functions on word-hyperbolic groups satisfy a central
limit theorem: if \bar{\phi}_n is the value of \phi on a random element of word
length n (in a certain sense), there are E and \sigma for which there is
convergence in the sense of distribution n^{-1/2}(\bar{\phi}_n - nE) \to
N(0,\sigma), where N(0,\sigma) denotes the normal distribution with standard
deviation \sigma. As a corollary, we show that if S_1 and S_2 are any two
finite generating sets for G, there is an algebraic number lambda_{1,2}
depending on S_1 and S_2 such that almost every word of length n in the S_1
metric has word length n\lambda_{1,2} in the S_2 metric, with error of size
O(\sqrt{n}).Comment: 26 pages; version 3: typos corrected, referee's comments incorporate
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