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Phase Transition in Subshifts of Finite Type via Hofbauer Potential

Shamsa Ishaq

TL;DR

The paper investigates freezing phase transitions for the Hofbauer potential defined on mixing subshifts of finite type, linking the pressure function to the entropy of the subshift and the concentration of the equilibrium measure post-transition. It employs an inducing-scheme approach à la Leplaideur, analyzing return-word decompositions into free and excursion components to control the key λ_{t,z} sums. The main result establishes a finite critical point t_c at which the equilibrium measure transitions from a unique, full-support measure with analytic pressure P(t) to the Parry measure with P(t) equal to the subshift entropy ξ for t>t_c. This work extends freezing-phase phenomena to SFTs and clarifies how the Parry measure emerges as the dominating equilibrium state after the transition, with the post-transition pressure matching the subshift entropy.

Abstract

The aim of this article is to establish freezing phase transition of the pressure function, considering the generalized Hofbauer potential φ, which is connected to the distance from subshift of finite type ΣF in the full shift Σ over finite alphabets. Our objective is to prove that the pressure function exhibits phase transition with the dominating measure is highly concentrated on ΣF after the transition. Moreover, the pressure function attains C1-smoothness when the transition occurs at 2.

Phase Transition in Subshifts of Finite Type via Hofbauer Potential

TL;DR

The paper investigates freezing phase transitions for the Hofbauer potential defined on mixing subshifts of finite type, linking the pressure function to the entropy of the subshift and the concentration of the equilibrium measure post-transition. It employs an inducing-scheme approach à la Leplaideur, analyzing return-word decompositions into free and excursion components to control the key λ_{t,z} sums. The main result establishes a finite critical point t_c at which the equilibrium measure transitions from a unique, full-support measure with analytic pressure P(t) to the Parry measure with P(t) equal to the subshift entropy ξ for t>t_c. This work extends freezing-phase phenomena to SFTs and clarifies how the Parry measure emerges as the dominating equilibrium state after the transition, with the post-transition pressure matching the subshift entropy.

Abstract

The aim of this article is to establish freezing phase transition of the pressure function, considering the generalized Hofbauer potential φ, which is connected to the distance from subshift of finite type ΣF in the full shift Σ over finite alphabets. Our objective is to prove that the pressure function exhibits phase transition with the dominating measure is highly concentrated on ΣF after the transition. Moreover, the pressure function attains C1-smoothness when the transition occurs at 2.
Paper Structure (4 sections, 3 theorems, 11 equations, 3 figures)

This paper contains 4 sections, 3 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions and Examples
  3. Setting, main theorem and Idea of the proof
  4. Proof of theorem \ref{['th-conj']}

Key Result

Theorem 1

Let $T$ be a positive matrix with order $n$, then the following holds: (i) $\rho>0$, and $\rho$ is algebraically simple eigen value of $T$. (ii) There is a positive unique real vector $\underline{u}$ such that $T\underline{u}=\rho\underline{u}$, and $||\underline{u}||=1$. (iii)There is a positive un

Figures (3)

  • Figure 1: Accidents in return word
  • Figure 2: Accidents in return word
  • Figure 3: Return word of ${\mathcal{T}}_{1}$, and ${\mathcal{T}}_{2}$ type

Theorems & Definitions (4)

  • Theorem 1: Perron
  • Theorem 2
  • Lemma 4.1
  • proof