Multiplicative functionals on ensembles of non-intersecting paths

Borodin, Alexei; Corwin, Ivan; Remenik, Daniel

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 28-58.

Le but de cet article est de développer une théorie autour des noyaux de la forme « intégrale de chemin » qui apparaissent dans l’étude des processus déterminantaux et des familles de chemins sans intersection. Notre premier résultat montre comment des déterminants avec de tels noyaux apparaissent naturellement dans l’étude du quotient de fonctions de partition et d’espérances de fonctionnelles pour des familles de chemins sans intersection sur des graphes avec des pondérations. Notre second résultat montre comment les déterminants de Fredholm avec des noyaux étendus (comme ceux que l’on trouve dans le cas du processus déterminantal ${Airy}_2$) sont égaux à des déterminants de Fredholm avec des noyaux de la forme « intégrale de chemin ». Nous montrons aussi comment ce second résultat s’applique à une grande variété d’exemples dont le mouvement Brownien stationnaire de Dyson, le processus ${Airy}_2$, le processus de Pearcey, les processus ${Airy}_1$ et ${Airy}_{2\to 1}$ ainsi que les processus de Markov sur les partitions reliées aux $z$-mesures.

The purpose of this article is to develop a theory behind the occurrence of “path-integral” kernels in the study of extended determinantal point processes and non-intersecting line ensembles. Our first result shows how determinants involving such kernels arise naturally in studying ratios of partition functions and expectations of multiplicative functionals for ensembles of non-intersecting paths on weighted graphs. Our second result shows how Fredholm determinants with extended kernels (as arise in the study of extended determinantal point processes such as the ${Airy}_2$ process) are equal to Fredholm determinants with path-integral kernels. We also show how the second result applies to a number of examples including the stationary (GUE) Dyson Brownian motion, the ${Airy}_2$ process, the Pearcey process, the ${Airy}_1$ and ${Airy}_{2\to 1}$ processes, and Markov processes on partitions related to the $z$-measures.

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