scholarly journals Hypergroups derived from random walks on some infinite graphs

2019 ◽  
Vol 189 (2) ◽  
pp. 321-353
Author(s):  
Tomohiro Ikkai ◽  
Yusuke Sawada
Keyword(s):  
Random Walks ◽  
Infinite Graphs

scholarly journals Average properties of combinatorial problems and thermodynamics of spin models on graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Alessandro Vezzani ◽  
Davide Cassi ◽  
Raffaella Burioni
Keyword(s):  
Random Walks ◽  
Spontaneous Magnetization ◽  
Discrete Symmetry ◽  
Combinatorial Problems ◽  
Infinite Graphs ◽  
Continuous Symmetry ◽  
Spin Models ◽  
Graph Topology ◽  
International Audience ◽  
Classical Spin Models

International audience The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spinmodels with O(n) continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized on graphs exhibiting percolation on the average. In this paper we define the combinatorial problems on the average, showing that they give rise to classifications of graph topology which are different from the ones obtained in usual (local) random-walks and percolation. Furthermore, we illustrate the theorem proving the correspondence between Potts model and average percolation.

scholarly journals Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

2017 ◽  
Vol 45 (4) ◽  
pp. 2655-2706 ◽  
Author(s):  
Codina Cotar ◽  
Debleena Thacker
Keyword(s):  
Random Walks ◽  
Infinite Graphs ◽  
Reinforced Random Walks

scholarly journals Discrete Random Walks on One-Sided ``Periodic'' Graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Generating Functions ◽  
Connected Graph ◽  
Infinite Graphs ◽  
Reflected Brownian Motion ◽  
Strongly Connected ◽  
International Audience ◽  
Null Recurrence

International audience In this paper we consider discrete random walks on infinite graphs that are generated by copying and shifting one finite (strongly connected) graph into one direction and connecting successive copies always in the same way. With help of generating functions it is shown that there are only three types for the asymptotic behaviour of the random walk. It either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence.

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