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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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.


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

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

First available in Project Euclid: 4 January 2011

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Zentralblatt MATH identifier

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

Determinantal point processes Random growth GUE minor process Pearcey process


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.

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