Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured
in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres conjecture in [3] relating sharp cut off, spectral gap, and mixing time.
This note gives a central limit theorem for the length of the longest subsequence of a random permutation which follows some repeating pattern. This includes the case of any fixed pattern of ups and downs which has at least one of each, such as the a
lternating case considered by Stanley in [2] and Widom in [3]. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutations.
A permutation $sigma$ describing the relative orders of the first $n$ iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is called an emph{order pattern}. For fixed $f$ and $n$, measuring the points $xin I$ (according to Lebesgue
measure) that generate the order pattern $sigma$ gives a probability distribution $mu_n(f)$ on the set of length $n$ permutations. We study the distributions that arise this way for various classes of functions $f$. Our main results treat the class of measure preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each $n$ this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general $f$, apart from an obvious compatibility condition, there is no restriction on the sequence ${mu_n(f)}$ for $n=1,2,...$. In addition, we give a necessary condition for $f$ to have emph{finite exclusion type}, i.e., for there to be finitely many order patterns that generate all order patterns not realized by $f$. Using entropy we show that if $f$ is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then $f$ cannot have finite exclusion type. This generalizes results of S. Elizalde.
