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Infinite-dimensional genetic and evolution algebras generated by Gibbs measures

Cristian F. Coletti, Lucas R. de Lima, Denis A. Luiz

Abstract

Genetic and evolution algebras arise naturally from applied probability and stochastic processes. Gibbs measures describe interacting systems commonly studied in thermodynamics and statistical mechanics with applications in several fields. Here, we consider that the algebras are determined by configurations of finite spins on a countable set with their associated Gibbs distributions. The model preserves properties of the finite-dimensional Gibbs algebras found in the literature and extend their results. We introduce infertility in the genetic dynamics when the configurations differ macroscopically. It induces a decomposition of the algebra into a direct sum of fertile ideals with genetic realization. The proposed infinite-dimensional algebras are commutative, non-associative, with uncountable basis and zero divisors. The properties of Gibbs measures allow us to deal with the difficulties arising from the algebraic structure and obtain the results presented in this article.

Infinite-dimensional genetic and evolution algebras generated by Gibbs measures

Abstract

Genetic and evolution algebras arise naturally from applied probability and stochastic processes. Gibbs measures describe interacting systems commonly studied in thermodynamics and statistical mechanics with applications in several fields. Here, we consider that the algebras are determined by configurations of finite spins on a countable set with their associated Gibbs distributions. The model preserves properties of the finite-dimensional Gibbs algebras found in the literature and extend their results. We introduce infertility in the genetic dynamics when the configurations differ macroscopically. It induces a decomposition of the algebra into a direct sum of fertile ideals with genetic realization. The proposed infinite-dimensional algebras are commutative, non-associative, with uncountable basis and zero divisors. The properties of Gibbs measures allow us to deal with the difficulties arising from the algebraic structure and obtain the results presented in this article.
Paper Structure (9 sections, 19 theorems, 112 equations, 4 figures)

This paper contains 9 sections, 19 theorems, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. On Gibbs measures
  3. On genetic algebras
  4. On evolution algebras
  5. Infinite-dimensional Gibbs genetic algebras
  6. Infinite-dimensional Gibbs evolution algebras
  7. Final remarks and open questions
  8. Declaration
  9. Acknowledgements

Key Result

Lemma 2.1

Let $\mathbb{L}$ and $S$ be finite. Consider $\Phi$ an admissible interaction potential and $\mu \in \mathscr{G}(\Phi)$ a Gibbs measure. Then, for every fixed set of clusters $\mathscr{C}$,

Figures (4)

  • Figure 1: Offspring alternating configurations in the clusters intersecting $\mathfrak{D}_{\sigma\eta}$ (the set of parental discrepancies).
  • Figure 2: Regions determined by \ref{['thm:finite.subalgebras']} that eliminate the boundary effect on finite subalgebras.
  • Figure 3: Self-similarity of the genealogical tree of $e_{\sigma\eta}$ when $\vert\Omega_{\sigma\eta}\vert=2$ and the stability of the idempotents.
  • Figure 4: Graphical representation of $f_{\sigma \eta}$ mapping configurations of $(\mathtt{E}^\sigma)^2$ to $(\mathtt{E}^\eta)^2$.

Theorems & Definitions (43)

  • Example 1: The Potts' model without external field
  • Example 2: Equivalent interaction potentials
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 3: Unique cluster
  • Example 4: Atomic clusters and two spin systems
  • Lemma 2.3
  • proof
  • ...and 33 more