Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limits of determinantal processes near a tacnode

Alexei Borodin and Maurice Duits

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Abstract

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.

Résumé

Nous étudions un processus de Markov sur un système de particules entrelacées. Lorsque le temps t est grand, les particules remplissent un domaine dépendant d’un paramètre ε > 0. Ce domaine possède deux points de rebroussement, dont l’un pointe vers le haut et l’autre vers le bas. À la limite ε ↓ 0, les deux points de rebroussement sont tangents, formant ainsi un tacnode. Le résultat principal de cet article est un calcul du noyau de corrélation locale autour du point tacnodal pendant le régime de transition ε ↓ 0. Nous démontrons aussi que le processus local interpole entre le processus de Pearcey et le processus des mineurs du GUE.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 243-258.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170237

Digital Object Identifier
doi:10.1214/10-AIHP373

Mathematical Reviews number (MathSciNet)
MR2779404

Zentralblatt MATH identifier
1208.82039

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes

Keywords
Determinantal point processes Random growth GUE minor process Pearcey process

Citation

Borodin, Alexei; Duits, Maurice. Limits of determinantal processes near a tacnode. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 243--258. doi:10.1214/10-AIHP373. https://projecteuclid.org/euclid.aihp/1294170237


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