Random walks on co-compact fuchsian groups

Sébastien Gouëzel; Steven P. Lalley

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 1, page 131-175
  • ISSN: 0012-9593

Abstract

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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 case, p n ( x , y ) ∼ C x , y R - n n - 3 / 2 .

How to cite

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Gouëzel, Sébastien, and Lalley, Steven P.. "Random walks on co-compact fuchsian groups." Annales scientifiques de l'École Normale Supérieure 46.1 (2013): 131-175. <http://eudml.org/doc/272109>.

@article{Gouëzel2013,
abstract = {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 case, $p^n(x,y)\sim C_\{x,y\}R^\{-n\}n^\{-3/2\}$.},
author = {Gouëzel, Sébastien, Lalley, Steven P.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {hyperbolic group; surface group; random walk; Green’s function; Gromov boundary; Martin boundary; Ruelle operator theorem; Gibbs state; local limit theorem},
language = {eng},
number = {1},
pages = {131-175},
publisher = {Société mathématique de France},
title = {Random walks on co-compact fuchsian groups},
url = {http://eudml.org/doc/272109},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Gouëzel, Sébastien
AU - Lalley, Steven P.
TI - Random walks on co-compact fuchsian groups
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 1
SP - 131
EP - 175
AB - 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 case, $p^n(x,y)\sim C_{x,y}R^{-n}n^{-3/2}$.
LA - eng
KW - hyperbolic group; surface group; random walk; Green’s function; Gromov boundary; Martin boundary; Ruelle operator theorem; Gibbs state; local limit theorem
UR - http://eudml.org/doc/272109
ER -

References

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