Fixed points of an infinite-dimensional operator related to Gibbs measures
Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 329-344
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We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set $\mathbb N$ of spin values on a Cayley tree. This operator is defined by a countable set of parameters $\lambda_i>0$, $a_{ij}\in\{0,1\}$, $i,j\in\mathbb N$. We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model.
Keywords:
fixed point, Cayley tree, Gibbs measure, HC model.
@article{TMF_2023_214_2_a11,
author = {U. R. Olimov and U. A. Rozikov},
title = {Fixed points of an infinite-dimensional operator related to {Gibbs} measures},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {329--344},
publisher = {mathdoc},
volume = {214},
number = {2},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/}
}
TY - JOUR AU - U. R. Olimov AU - U. A. Rozikov TI - Fixed points of an infinite-dimensional operator related to Gibbs measures JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2023 SP - 329 EP - 344 VL - 214 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/ LA - ru ID - TMF_2023_214_2_a11 ER -
U. R. Olimov; U. A. Rozikov. Fixed points of an infinite-dimensional operator related to Gibbs measures. Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 329-344. http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a11/