Even for these graphs, the asymptotic behavior can be rather complex. We also carry out a fairly detailed analysis of the stationary distribution of this process for several simple classes graphs, such as paths and cycles. For example, in the uniform case (where p = 1), for each subset T of vertices of G there is an eigenvalue 2 λT (with multiplicity 1) which is just equal to the number of edges in the subgraph induced by T divided by the number of edges of G. We show that the eigenvalues for this random walk can be naturally indexed by subsets of the vertices of G. This “edge flipping ” process generates a random walk on the set of all possible color patterns on G. At each step in our process, we select a random edge of G and (re-)color both its endpoints blue with probability p, or red with probability a = 1 − p. Suppose we begin with some finite graph G in which each vertex of G is initially arbitrarily colored red or blue. A specific example of what we study is the following. These processes are related to the so-called Tsetlin library random walk as well as to some variants of the classical voter model. In this paper we investigate certain random processes on graphs first suggested by Persi Diaconis. When p = 2, spectral theory allows for a deeper analysis of the cutoff phenomenon producing in some cases the asymptotic behavior of the sequences (tn) Ideally, when a cutoff exists, we would like to determine precisely tn and bn. The notion of cutoff for a family of Markov chains indexed by n involves a cutoff time sequence (tn) 1 and window size sequence (bn) 1. One of the main result of the thesis is that for families of reversible Markov chains and 1 < p ≤ ∞, the existence of an `p-cutoff can be characterized using two parameters: the spectral gap and the mixing time. For p = 1, one recovers the classical total variation distance. We focus on the case when the convergence is measured at the `p-distance for 1 ≤ p ≤ ∞. Our aim is to develop a theory of this phenomenon and to illustrate this theory with interesting examples. For these models, after a waiting period, the chain abruptly converges to its stationary distribution. In this dissertation, we discuss a behavior -the cutoff phenomenon - that is known to appear in many models. Similar convergence rate questions for finite Markov chains are important in many fields including statistical physics, computer science, biology and more. Below in the introduction we recall some well known facts about Coxeter groups, weak Bruhat order, and Poincar'e seri.A card player may ask the following question: how many shuffles are needed to mix up a deck of cards? Mathematically, this question falls in the realm of the quantitative study of the convergence of finite Markov chains. Some of the simplest pictures of the labeled Hasse diagrams in Section 1 have appeared also in connection with Verma modules and Schubert cells and as a graphical device for calculating the homology of the most elementary Artin groups. ) - and by (2) deriving simple non-recursive schemes for the computation of standard reduced words for both unsigned and signed permutations.
#INTERSECTION OF A LATTICE WITH HYPERPLAN SERIES#
In the present paper we show the usefulness of this partition property by (1) giving a pictorial combinatorial derivation of the Poincar'e polynomials and series for the finite irreducible Coxeter groups and the affine Coxeter groups on three generators - results, which until now have been obtained by invariant theoretic or Lie theoretic methods (cf. For 3-simplex, any plane within the tetrahedron (where there are 6. But the partitioning property of the weak Bruhat order of Coxeter groups into isomorphic parts as stated in Theorem 0.1 below - though probably known by the experts - has certainly not been fully exploited. If we contract a (n-1)d hyperplane with a n-simplex, then what is maximum number of intersection points with the egdes of the simplex and the hyperplane For, if we draw a line within a 2-simplex (there are 3 edges), it will have a intersection of maximum two edges. The combinatorial properties of weak Bruhat order of Coxeter groups, especially of the finite irreducible and affine ones, have been investigated for quite a time (see for example and the references therein). The algebraic basis for both (1) and (2) is a simple partition property of the weak Bruhat order of Coxeter groups into isomorphic parts. (2) Non-recursive methods for the computation of `standard reduced words' for (signed) permutations are described.
The Poincar'e polynomials of the finite irreducible Coxeter groups and the Poincar'e series of the affine Coxeter groups on three generators are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory.
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