scholarly journals A Numerical Lower Bound for the Spectral Radius of Random Walks on Surface Groups

2015 ◽  
Vol 24 (6) ◽  
pp. 838-856 ◽  
Author(s):  
S. GOUEZEL
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Spectral Radius ◽  
Cayley Graphs ◽  
Surface Group ◽  
Hyperbolic Groups ◽  
Surface Groups ◽  
Genus 2 ◽  
Order Of Magnitude

Estimating numerically the spectral radius of a random walk on a non-amenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.

scholarly journals Non-intersection of transient branching random walks

2020 ◽  
Vol 178 (1-2) ◽  
pp. 1-23
Author(s):  
Tom Hutchcroft
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Cayley Graph ◽  
Spectral Radius ◽  
Branching Random Walk ◽  
Expected Number ◽  
Branching Random Walks ◽  
Offspring Distribution ◽  
The Future ◽  
Supercritical Phase

Abstract Let G be a Cayley graph of a nonamenable group with spectral radius $$\rho < 1$$ ρ < 1 . It is known that branching random walk on G with offspring distribution $$\mu $$ μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $${\overline{\mu }}$$ μ ¯ satisfies $$\overline{\mu }\le \rho ^{-1}$$ μ ¯ ≤ ρ - 1 . Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase $$1<\overline{\mu } \le \rho ^{-1}$$ 1 < μ ¯ ≤ ρ - 1 , and in particular at the recurrence threshold $${\overline{\mu }} = \rho ^{-1}$$ μ ¯ = ρ - 1 , the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.

scholarly journals Surface groups within Baumslag doubles

2011 ◽  
Vol 54 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Surface Groups ◽  
Word Hyperbolic Group ◽  
Genus 2

AbstractA conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

scholarly journals Random walks on weakly hyperbolic groups

2018 ◽  
Vol 2018 (742) ◽  
pp. 187-239 ◽  
Author(s):  
Joseph Maher ◽  
Giulio Tiozzo
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Linear Growth ◽  
Countable Group ◽  
Convergence Result ◽  
Hyperbolic Groups ◽  
Poisson Boundary ◽  
Gromov Hyperbolic ◽  
Gromov Boundary ◽  
Translation Length

Abstract Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

On the range of the transient frog model on ℤ

2017 ◽  
Vol 49 (2) ◽  
pp. 327-343 ◽  
Author(s):  
Arka Ghosh ◽  
Steven Noren ◽  
Alexander Roitershtein
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Explicit Formula ◽  
Infinite System ◽  
Scaling Limits ◽  
Interacting Random Walks ◽  
Degenerate Distribution ◽  
Frog Model ◽  
Asymptotic Scaling

Abstract We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

Commensurability classes of arithmetic Fuchsian surface groups of genus 2

2009 ◽  
Vol 148 (1) ◽  
pp. 117-133
Author(s):  
C. MACLACHLAN ◽  
G. ROSENBERGER
Keyword(s):  
Real Number ◽  
Number Field ◽  
Quaternion Algebra ◽  
Surface Group ◽  
Fuchsian Groups ◽  
Surface Groups ◽  
Genus 2 ◽  
Totally Real ◽  
Real Number Field

AbstractHere we determine the arithmetic data i.e. the totally real number field and the set of ramified places of the defining quaternion algebra, of all those commensurability classes of arithmetic Fuchsian groups which contain a surface group of genus 2, i.e. a group of signature (2;– –).

scholarly journals Renewal theory for random walks on surface groups

2016 ◽  
Vol 38 (1) ◽  
pp. 155-179 ◽  
Author(s):  
PETER HAÏSSINSKY ◽  
PIERRE MATHIEU ◽  
SEBASTIAN MÜLLER
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Asymptotic Variance ◽  
Renewal Theory ◽  
Surface Groups ◽  
Exponential Moment ◽  
Moment Conditions

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as analigned unionof independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

scholarly journals CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS

2021 ◽  
pp. 1-46
Author(s):  
Charles Bordenave ◽  
Hubert Lacoin
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Mixing Time ◽  
Simple Random Walk ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Expander Graphs ◽  
Transitive Graph ◽  
Regular Tree

Abstract It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T} _d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound. It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

scholarly journals A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

2008 ◽  
Author(s):  
Sebastian Cioaba ◽  
Edwin van Dam ◽  
Jack Koolen ◽  
Jae-Ho Lee
Keyword(s):  
Lower Bound ◽  
Spectral Radius

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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