scholarly journals Occupation measure of random walks and wired spanning forests in balls of Cayley graphs

2020 ◽  
Vol 29 (1) ◽  
pp. 97-109
Author(s):  
Russell Lyons ◽  
Yuval Peres ◽  
Xin Sun ◽  
Tianyi Zheng
Keyword(s):  
Random Walks ◽  
Cayley Graphs ◽  
Occupation Measure ◽  
Spanning Forests

scholarly journals Random walks on Cayley graphs of complex reflection groups

2021 ◽  
pp. 51-56
Author(s):  
Maksim M. Vaskouski
Keyword(s):  
Random Walks ◽  
Cayley Graphs ◽  
Asymptotic Properties ◽  
Expander Graphs ◽  
Spectral Gaps ◽  
Reflection Groups ◽  
Logarithmic Order ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Special Case

Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

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.

Non-homogeneous Random Walks

2016 ◽  
Author(s):  
Mikhail Menshikov ◽  
Serguei Popov ◽  
Andrew Wade
Keyword(s):  
Random Walks
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