scholarly journals Word Metric, Stationary Measure and Minkowski’s Question Mark Function

2020 ◽  
Vol 15 (2) ◽  
pp. 23-38
Author(s):  
Uriya Pumerantz
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Limit Point ◽  
Projective Line ◽  
Specific Measure ◽  
Question Mark ◽  
Convolution Power ◽  
Countably Infinite

AbstractGiven a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x∈ X and a function f over X we consider the sums Sn(f, x) = ∑g∈Gnf(gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (03) ◽  
pp. 640-654
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres K n of some volume V(K n ) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form L n (s) molecules with exactly s atoms K n. The random variable L n(s) has a limit distribution if V(K n ) tends to zero but nV(Kn ) tends to infinity at a certain rate: it is shown that for L n(s) is asymptotically Poisson.

A Lower Estimate for Central Probabilities on Polycyclic Groups

1992 ◽  
Vol 44 (5) ◽  
pp. 897-910 ◽  
Author(s):  
G. Alexopoulos
Keyword(s):  
Probability Measure ◽  
Lie Group ◽  
Exponential Growth ◽  
Lower Estimate ◽  
Large Time ◽  
Polycyclic Group ◽  
Central Value ◽  
Convolution Power ◽  
Polycyclic Groups

AbstractWe give a lower estimate for the central value μ*n(e) of the nth convolution power μ*···*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.

scholarly journals The probabilities of extinction in a branching random walk on a strip

2020 ◽  
Vol 57 (3) ◽  
pp. 811-831
Author(s):  
Peter Braunsteins ◽  
Sophie Hautphenne
Keyword(s):  
Random Walk ◽  
Iterative Method ◽  
Fixed Points ◽  
Branching Processes ◽  
Extinction Probability ◽  
Branching Random Walk ◽  
Infinite Type ◽  
Hessenberg Form ◽  
Global Extinction ◽  
Countably Infinite

AbstractWe consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

scholarly journals Random spanning trees of Cayley graphs and an associated compactification of semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 611-620
Author(s):  
Steven N. Evans
Keyword(s):  
Random Walk ◽  
Cayley Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finite Tree ◽  
Generating Set ◽  
Countably Infinite

A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

Asymptotic distribution theory for Hoare's selection algorithm

1996 ◽  
Vol 28 (1) ◽  
pp. 252-269 ◽  
Author(s):  
Rudolf Grübel ◽  
Uwe Rösler
Keyword(s):  
Stochastic Process ◽  
Random Walk ◽  
Asymptotic Behaviour ◽  
Asymptotic Distribution ◽  
Random Environment ◽  
Binary Tree ◽  
Selection Algorithm ◽  
Real Numbers ◽  
Asymptotic Distribution Theory ◽  
Path Lengths

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

"MÜNCHHAUSEN TRICK" AND AMENABILITY OF SELF-SIMILAR GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 907-937 ◽  
Author(s):  
VADIM A. KAIMANOVICH
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Group Identity ◽  
Convex Combination ◽  
Similar Group ◽  
Grigorchuk Group ◽  
Intermediate Growth ◽  
Group A ◽  
Self Similar ◽  
Asymptotic Entropy

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μw. If the measure μ is self-similar in the sense that μw is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (3) ◽  
pp. 640-654 ◽  
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for Ln(s) is asymptotically Poisson.

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