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Restricted Permutations and Permanents of Infinite Amenable Groups

Hanfeng Li, Klaus Schmidt

TL;DR

This work studies restricted-permutation shifts X_A over countable amenable groups Γ, showing that their dynamics can be analyzed via local pressure and entropy formulas. It introduces permanents per(f) for nonnegative elements of the real group algebra and connects them to topological pressure via P(X_A, log f), while comparing to the Fuglede-Kadison determinant; the results illuminate when these permanents encode entropy and when determinants provide analogous invariants. The authors develop finite-quotient approximations, establish equality of entropies for X_A and its injective variant X_A^ι, and provide general bounds and special-case identities (e.g., dimers) that tie combinatorial matrix properties to dynamical quantities. Overall, the paper bridges symbolic dynamics of restricted permutations with operator-algebraic and combinatorial tools, offering both conceptual insights and tractable bounds in a broad amenable-group setting.

Abstract

Let $Γ$ be an infinite discrete group and $\mathsf{A}\subset Γ$ a nonempty finite subset. The set of permutations $σ$ of $Γ$ such that $s^{-1}σ(s)\in \mathsf{A}$ for every $s\in Γ$ can be identified with a shift of finite type $X_\mathsf{A}\subset \mathsf{A}^Γ$ over $Γ$. In this paper we study dynamical properties of such shift spaces, like invariant probability measures, topological entropy, and topological pressure, under the hypothesis that $Γ$ is amenable. In this case the topological entropy $\textrm{h}_{\textrm{top}}(X_\mathsf{A})$ can be expressed as logarithmic growth rate of permanents of certain finite (0,1)-matrices associated with right Følner sequences in $Γ$. Motivated by the difficulty of computing such permanents we introduce the notion of the permanent $\textrm{per}(f)$ for nonnegative elements $f$ in the real group ring $\mathbb{R}Γ$ of $Γ$ whose support is the alphabet $\mathsf{A}$ of the shift space $X_\mathsf{A}$, and compare, for arbitrary $f \in \mathbb{R}Γ$, the Fuglede-Kadison determinant $\textrm{det} _\textrm{FK}(f)$ with the permanent $\textrm{per}(|f|)$ of the absolute value $|f|$ of $f$. Although this approach is effective in only few examples, discussed below, it is interesting from a conceptual point of view that the permanent $\textrm{per}(f)$ of a nonnegative element $f\in \mathbb{R}Γ$ can be viewed as topological pressure of the restricted-permutation shift space $X_\mathsf{A}$ associated with the function $\log f$ on the alphabet $\mathsf{A}=\textrm{supp}(f)$ of $X_\mathsf{A}$.

Restricted Permutations and Permanents of Infinite Amenable Groups

TL;DR

This work studies restricted-permutation shifts X_A over countable amenable groups Γ, showing that their dynamics can be analyzed via local pressure and entropy formulas. It introduces permanents per(f) for nonnegative elements of the real group algebra and connects them to topological pressure via P(X_A, log f), while comparing to the Fuglede-Kadison determinant; the results illuminate when these permanents encode entropy and when determinants provide analogous invariants. The authors develop finite-quotient approximations, establish equality of entropies for X_A and its injective variant X_A^ι, and provide general bounds and special-case identities (e.g., dimers) that tie combinatorial matrix properties to dynamical quantities. Overall, the paper bridges symbolic dynamics of restricted permutations with operator-algebraic and combinatorial tools, offering both conceptual insights and tractable bounds in a broad amenable-group setting.

Abstract

Let be an infinite discrete group and a nonempty finite subset. The set of permutations of such that for every can be identified with a shift of finite type over . In this paper we study dynamical properties of such shift spaces, like invariant probability measures, topological entropy, and topological pressure, under the hypothesis that is amenable. In this case the topological entropy can be expressed as logarithmic growth rate of permanents of certain finite (0,1)-matrices associated with right Følner sequences in . Motivated by the difficulty of computing such permanents we introduce the notion of the permanent for nonnegative elements in the real group ring of whose support is the alphabet of the shift space , and compare, for arbitrary , the Fuglede-Kadison determinant with the permanent of the absolute value of . Although this approach is effective in only few examples, discussed below, it is interesting from a conceptual point of view that the permanent of a nonnegative element can be viewed as topological pressure of the restricted-permutation shift space associated with the function on the alphabet of .
Paper Structure (10 sections, 37 theorems, 240 equations)

This paper contains 10 sections, 37 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. The local pressure formula
  3. The local pressure formula for shift actions
  4. The local entropy formula for algebraic actions
  5. The local pressure formula for subshifts
  6. Local entropy for restricted permutations
  7. Permanents for elements of the group algebra
  8. Permanent vs Determinant
  9. Approximation by finite quotient groups
  10. Bounds for permanents

Key Result

Lemma 2.1

Let $f\in C(\mathsf{B})$ and let $\mathscr{U}$ be a finite open cover of $\mathsf{B}$. As $F\in \mathscr{F}(\Gamma )$ becomes more and more right invariant, the limits $\lim_F \frac{1}{|F|}\log p_F(f, \pi^{-1}(\mathscr{U}), X_\mathcal{P})$ and $\lim_F \frac{1}{|F|}\log p_F(f, \mathscr{U}, X_{\mathca

Theorems & Definitions (79)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 69 more