852 research outputs found
Directed nonabelian sandpile models on trees
We define two general classes of nonabelian sandpile models on directed trees
(or arborescences) as models of nonequilibrium statistical phenomena. These
models have the property that sand grains can enter only through specified
reservoirs, unlike the well-known abelian sandpile model.
In the Trickle-down sandpile model, sand grains are allowed to move one at a
time. For this model, we show that the stationary distribution is of product
form. In the Landslide sandpile model, all the grains at a vertex topple at
once, and here we prove formulas for all eigenvalues, their multiplicities, and
the rate of convergence to stationarity. The proofs use wreath products and the
representation theory of monoids.Comment: 43 pages, 5 figures; introduction improve
Markov chains, -trivial monoids and representation theory
We develop a general theory of Markov chains realizable as random walks on
-trivial monoids. It provides explicit and simple formulas for the
eigenvalues of the transition matrix, for multiplicities of the eigenvalues via
M\"obius inversion along a lattice, a condition for diagonalizability of the
transition matrix and some techniques for bounding the mixing time. In
addition, we discuss several examples, such as Toom-Tsetlin models, an exchange
walk for finite Coxeter groups, as well as examples previously studied by the
authors, such as nonabelian sandpile models and the promotion Markov chain on
posets. Many of these examples can be viewed as random walks on quotients of
free tree monoids, a new class of monoids whose combinatorics we develop.Comment: Dedicated to Stuart Margolis on the occasion of his sixtieth
birthday; 71 pages; final version to appear in IJA
Submonoids and rational subsets of groups with infinitely many ends
In this paper we show that the membership problems for finitely generated
submonoids and for rational subsets are recursively equivalent for groups with
two or more ends
Automaton semigroup constructions
The aim of this paper is to investigate whether the class of automaton
semigroups is closed under certain semigroup constructions. We prove that the
free product of two automaton semigroups that contain left identities is again
an automaton semigroup. We also show that the class of automaton semigroups is
closed under the combined operation of 'free product followed by adjoining an
identity'. We present an example of a free product of finite semigroups that we
conjecture is not an automaton semigroup. Turning to wreath products, we
consider two slight generalizations of the concept of an automaton semigroup,
and show that a wreath product of an automaton monoid and a finite monoid
arises as a generalized automaton semigroup in both senses. We also suggest a
potential counterexample that would show that a wreath product of an automaton
monoid and a finite monoid is not a necessarily an automaton monoid in the
usual sense.Comment: 13 pages; 2 figure
Automaton semigroups: new construction results and examples of non-automaton semigroups
This paper studies the class of automaton semigroups from two perspectives:
closure under constructions, and examples of semigroups that are not automaton
semigroups. We prove that (semigroup) free products of finite semigroups always
arise as automaton semigroups, and that the class of automaton monoids is
closed under forming wreath products with finite monoids. We also consider
closure under certain kinds of Rees matrix constructions, strong semilattices,
and small extensions. Finally, we prove that no subsemigroup of arises as an automaton semigroup. (Previously, itself was
the unique example of a finitely generated residually finite semigroup that was
known not to arise as an automaton semigroup.)Comment: 27 pages, 6 figures; substantially revise
Left-Garside categories, self-distributivity, and braids
In connection with the emerging theory of Garside categories, we develop the
notions of a left-Garside category and of a locally left-Garside monoid. In
this framework, the connection between the self-distributivity law LD and
braids amounts to the result that a certain category associated with LD is a
left-Garside category, which projects onto the standard Garside category of
braids. This approach leads to a realistic program for establishing the
Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhauser
(2000), Chap. IX]
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain
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