Description of fixed points of an infinite dimensional operator

TL;DR

This paper analyzes fixed points of an infinite-dimensional non-linear operator related to a hard core model by reducing it to a two-dimensional problem, explicitly finding up to seven fixed points based on parameter subsets.

Contribution

It introduces a novel reduction technique from infinite to two dimensions to fully analyze fixed points of a complex operator.

Findings

Number of fixed points can be up to seven.

Explicit forms of all fixed points are derived.

Fixed points are classified based on their location relative to the line y=x.

Abstract

We consider an infinite-dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values. It is known that finding the fixed points of an infinite-dimensional operator is generally impossible. But we have fully analyzed the fixed points of an infinite-dimensional operator by applying a technique of reducing an infinite-dimensional operator to a two-dimensional operator. The set of parameters is divided into subsets $A_{i,j},$ where the index $i$ means the number of fixed points on the line $y=x$, $j$ means the number of fixed points outside of $y=x.$ The number of fixed points can be up to seven, and the explicit form of each fixed point is found.