TY - JOUR
T1 - Maximum of the Membrane Model on Regular Trees
AU - Cipriani, Alessandra
AU - Dan, Biltu
AU - Hazra, Rajat Subhra
AU - Ray, Rounak
PY - 2023
Y1 - 2023
N2 - The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over Zd, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei in (Ann Probab 12:311–314, 1984). We exploit this representation on m-regular trees and show that the infinite volume limit on the infinite tree exists when m≥ 3. Further we determine the behavior of the maximum under the infinite and finite volume measures.
AB - The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over Zd, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei in (Ann Probab 12:311–314, 1984). We exploit this representation on m-regular trees and show that the infinite volume limit on the infinite tree exists when m≥ 3. Further we determine the behavior of the maximum under the infinite and finite volume measures.
KW - Extremes
KW - Membrane model
KW - Random interfaces
KW - Random walk representation
KW - Trees
UR - http://www.scopus.com/inward/record.url?scp=85142885578&partnerID=8YFLogxK
U2 - 10.1007/s10955-022-03043-w
DO - 10.1007/s10955-022-03043-w
M3 - Article
AN - SCOPUS:85142885578
SN - 0022-4715
VL - 190
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 1
M1 - 25
ER -