Random walks on finite rank solvable groups

Ch. Pittet; Laurent Saloff-Coste

Displaying similar documents to “Random walks on finite rank solvable groups”

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

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We show that there exists a finitely generated group of growth $\sim f$ for all functions $f : ℝ_{+} \to ℝ_{+}$ satisfying $f (2 R) \leq f {(R)}^{2} \leq f (η_{+} R)$ for all $R$ large enough and $η_{+} \approx 2.4675$ the positive root of $X^{3} - X^{2} - 2 X - 4$ . Set $α_{-} = log 2 / log η_{+} \approx 0.7674$ ; then all functions that grow uniformly faster than $exp (R^{α_{-}})$ are realizable as the growth of a group. We also give a family of sum-contracting branched groups of growth $\sim exp (R^{α})$ for a dense set of $α \in [α_{-}, 1]$ .

The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

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We show that, given an n-dimensional normed space X, a sequence of $N = {(8 / ε)}^{2 n}$ independent random vectors ${(X_{i})}_{i = 1}^{N}$ , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ : ℝ \to ℝ^{N}$ defined by $Γ x = {(⟨ x, X_{i} ⟩)}_{i = 1}^{N}$ embeds X in $ℓ_{\infty}^{N}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{\infty}^{N}$ with asymptotically best possible relation between N, n, and ε.

About the generating function of a left bounded integer-valued random variable

Charles Delorme, Jean-Marc Rinkel (2008)

Bulletin de la Société Mathématique de France

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We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1 - Φ (z)$ inside the unit disk, where $Φ$ is the generating function of $X$ , under some mild conditions

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

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We study a continuous time growth process on the $d$ -dimensional hypercubic lattice $𝒵^{d}$ , which admits a phenomenological interpretation as the combustion reaction $A + B \to 2 A$ , where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

Random walks on co-compact fuchsian groups

Sébastien Gouëzel, Steven P. Lalley (2013)

Annales scientifiques de l'École Normale Supérieure

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It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$ . It is also shown that Ancona’s inequalities extend to $R$ , and therefore that the Martin boundary for $R$ -potentials coincides with the natural geometric boundary $S^{1}$ , and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic...

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

Positivity of integrated random walks

Vladislav Vysotsky (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n} : = \sum_{i = 1}^{n} S_{i}$ . Suppose $S_{1}$ is in the domain of normal attraction of an $α$ -stable law with $1 l t; α \leq 2$ . Assuming that $S_{1}$ is either right-exponential (i.e. $ℙ (S_{1} g t; x | S_{1} g t; 0) = e^{- a x}$ for some $a g t; 0$ and all $x g t; 0$ ) or right-continuous (skip free), we prove that $ℙ {A_{1} g t; 0, \dots, A_{N} g t; 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ as $N \to \infty$ , where $C_{α} g t; 0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Random walks in ${(ℤ_{+})}^{2}$ with non-zero drift absorbed at the axes

Irina Kurkova, Kilian Raschel (2011)

Bulletin de la Société Mathématique de France

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Spatially homogeneous random walks in ${(ℤ_{+})}^{2}$ with non-zero jump probabilities at distance at most $1$ , with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes. ...

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

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We consider random walk on a discrete torus $E$ of side-length $N$ , in sufficiently high dimension $d$ . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u N^{d}$ . We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $log N$ . Moreover, this connected component occupies a...

The spread of a catalytic branching random walk

Philippe Carmona, Yueyun Hu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a catalytic branching random walk on $ℤ$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_{n}$ : For some constant $α$ , $\frac{M_{n}}{n} \to α$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_{n} - α n$ as $n$ goes to infinity.

Expansion in $S L_{d} (𝒪_{K} / I)$ , $I$ square-free

Péter P. Varjú (2012)

Journal of the European Mathematical Society

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Let $S$ be a fixed symmetric finite subset of $S L_{d} (𝒪_{K})$ that generates a Zariski dense subgroup of $S L_{d} (𝒪_{K})$ when we consider it as an algebraic group over $m a t h b b Q$ by restriction of scalars. We prove that the Cayley graphs of $S L_{d} (𝒪_{K} / I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of $𝒪_{K}$ if $d = 2$ and $K$ is an arbitrary numberfield, or if $d = 3$ and $K = ℚ$ .

A note on normal generation and generation of groups

Andreas Thom (2015)

Communications in Mathematics

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In this note we study sets of normal generators of finitely presented residually $p$ -finite groups. We show that if an infinite, finitely presented, residually $p$ -finite group $G$ is normally generated by $g_{1}, \dots, g_{k}$ with order $n_{1}, \dots, n_{k} \in {1, 2, \dots} \cup {\infty}$ , then $β_{1}^{(2)} (G) \leq k - 1 - \sum_{i = 1}^{k} \frac{1}{n_{i}},$ where $β_{1}^{(2)} (G)$ denotes the first $ℓ^{2}$ -Betti number of $G$ . We also show that any $k$ -generated group with $β_{1}^{(2)} (G) \geq k - 1 - ε$ must have girth greater than or equal $1 / ε$ .

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

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Let ${X_{n, j}, 1 \leq j \leq m (n), n \geq 1}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0 < b_{n} \to \infty$ . Conditions are given for $\sum_{j = 1}^{m (n)} X_{n, j} / b_{n} \to 0$ completely and for ${max}_{1 \leq k \leq m (n)} | \sum_{j = 1}^{k} X_{n, j} | / b_{n} \to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum_{j = 1}^{m (n)} X_{n, j}^{*}$ of the dependent bootstrap samples ${{X_{n, j}^{*}, 1 \leq j \leq m (n)}, n \geq 1}$ arising from a sequence of i.i.d. random variables ${X_{n}, n \geq 1}$ .

A Non Solvable Operator Satisfying Condition $(ψ)$

Nicolas Lerner (1992)

Publications mathématiques et informatique de Rennes

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Factorization of CP-rank- $3$ completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

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A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A = B B^{⊤}$ . If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$ . In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$ . Failure of this algorithm implies that $A$ does not have cp-rank $3$ . Our motivation stems from the question if there exist three nonnegative polynomials of degree at...

On dicyclic groups as inner mapping groups of finite loops

Emma Leppälä, Markku Niemenmaa (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group with a dicyclic subgroup $H$ . We show that if there exist $H$ -connected transversals in $G$ , then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I (Q)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I (Q)$ is a certain type of a direct product.