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Nilpotent adjacency matrices, random graphs and quantum random variables

Identifieur interne : 004567 ( Main/Merge ); précédent : 004566; suivant : 004568

Nilpotent adjacency matrices, random graphs and quantum random variables

Auteurs : René Schott [France] ; George Stacey Staples [États-Unis]

Source :

Journal of physics. A, Mathematical and theoretical : (Print) [ 1751-8113 ] ; 2008.

RBID : Pascal:08-0216287

Descripteurs français

Pascal (Inist)
- Matrice aléatoire, Graphe aléatoire, Graphe quantique, Variable aléatoire, Théorie quantique, Probabilité, Graphe fini, Fermion, Opérateur annihilation, Dimension infinie, Vertex, Chaîne Markov.

English descriptors

KwdEn :
- Annihilation operators, Fermions, Finite graph, Infinite dimension, Markov chain, Probability, Quantum graph, Quantum theory, Random graph, Random matrix, Random variable, Vertex.

Abstract

While a number of researchers have previously investigated the relationship between graph theory and quantum probability, the current work explores a new perspective. The approach of this paper is to begin with an arbitrary graph having no previously established relationship to quantum probability and to use that graph to construct a quantum probability space in which moments of quantum random variables reveal information about the graph's structure. Given an arbitrary finite graph and arbitrary odd integer m≥ 3, fermion annihilation operators are used to construct a family of quantum random variables whose mth moments correspond to the graph's m-cycles. The approach is then generalized to recover a graph's m-cycles for any integer m≥ 3 by defining nilpotent adjacency operators in terms of null-square generators of an infinite-dimensional Abelian algebra. It is shown that ordering the vertices of a simple graph induces a canonical decompositionΨ =Ψ⁺ + Ψ^- on any nilpotent adjacency operator Ψ. The work concludes with applications to Markov chains and random graphs.

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Pascal:08-0216287

Le document en format XML

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<term>Probability</term>
<term>Quantum graph</term>
<term>Quantum theory</term>
<term>Random graph</term>
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<term>Graphe aléatoire</term>
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<term>Théorie quantique</term>
<term>Probabilité</term>
<term>Graphe fini</term>
<term>Fermion</term>
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<front><div type="abstract" xml:lang="en">While a number of researchers have previously investigated the relationship between graph theory and quantum probability, the current work explores a new perspective. The approach of this paper is to begin with an arbitrary graph having no previously established relationship to quantum probability and to use that graph to construct a quantum probability space in which moments of quantum random variables reveal information about the graph's structure. Given an arbitrary finite graph and arbitrary odd integer m≥ 3, fermion annihilation operators are used to construct a family of quantum random variables whose mth moments correspond to the graph's m-cycles. The approach is then generalized to recover a graph's m-cycles for any integer m≥ 3 by defining nilpotent adjacency operators in terms of null-square generators of an infinite-dimensional Abelian algebra. It is shown that ordering the vertices of a simple graph induces a canonical decompositionΨ =Ψ<sup>+</sup>
 + Ψ<sup>-</sup>
 on any nilpotent adjacency operator Ψ. The work concludes with applications to Markov chains and random graphs.</div>
</front>
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<country name="États-Unis"><noRegion><name sortKey="Stacey Staples, George" sort="Stacey Staples, George" uniqKey="Stacey Staples G" first="George" last="Stacey Staples">George Stacey Staples</name>
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Nilpotent adjacency matrices, random graphs and quantum random variables

Nilpotent adjacency matrices, random graphs and quantum random variables

Source :

Descripteurs français

English descriptors

Abstract

Links toward previous steps (curation, corpus...)

Links to Exploration step

Le document en format XML

Pour manipuler ce document sous Unix (Dilib)

Pour mettre un lien sur cette page dans le réseau Wicri