The Liouville property and Hilbertian compression

Gournay, Antoine

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2435-2454.

Des bornes inférieures sur l’exposant de compression hilbertienne équivariante $\alpha$ sont données en utilisant les marches aléatoires. Plus précisément, si la probabilité de retour de la marche aléatoire est $⪰\exp (-n^{\gamma })$ pour un graphe de Cayley, alors $\alpha \ge (1-\gamma )/(1+\gamma )$. Ceci motive l’étude de relations supplémentaires entre la probabilité de retour, la vitesse, l’entropie et la croissance du volume. Par exemple, if $\left|B_n\right|⪯e^{n^{\nu }}$, alors l’exposant de vitesse est $\le 1/(2-\nu )$.

Avec une hypothèse plus forte sur le comportement du noyau de la chaleur hors de la diagonale, la borne inférieure sur la compression $\alpha \ge 1-\gamma$. Par un résultat de Naor et Peres sur la compression et la vitesse des marches aléatoires, ceci donne un estimé prometteur sur la vitesse et implique la propriété de Liouville si $\gamma <1/2$.

Lower bounds on the equivariant Hilbertian compression exponent $\alpha$ are obtained using random walks. More precisely, if the probability of return of the simple random walk is $⪰\exp (-n^{\gamma })$ in a Cayley graph then $\alpha \ge (1-\gamma )/(1+\gamma )$. This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if $\left|B_n\right|⪯e^{n^{\nu }}$ then the speed exponent is $\le 1/(2-\nu )$.

Under a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to $\alpha \ge 1-\gamma$. Using a result from Naor & Peres on compression and the speed of random walks, this yields very promising bounds on speed and implies the Liouville property if $\gamma <1/2$.

Classification : 20F69, 60J50, 60G50, 37A35, 43A65
Keywords: Hilbertian compression, random walks on groups, entropy, drift, growth of groups
Mot clés : compression hilbertienne, marches aléatoires sur les groupes, entropie, vitesse, croissance des groupes

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