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), p. 28-58 / Harvested from Numdam

Résumé

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 $\operatorname{Airy}_{2}$) 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 $\operatorname{Airy}_{2}$, le processus de Pearcey, les processus $\operatorname{Airy}_{1}$ et $\operatorname{Airy}_{2\rightarrow 1}$ ainsi que les processus de Markov sur les partitions reliées aux $z$-mesures.

Abstract

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 $\operatorname{Airy}_{2}$ 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 $\operatorname{Airy}_{2}$ process, the Pearcey process, the $\operatorname{Airy}_{1}$ and $\operatorname{Airy}_{2\rightarrow 1}$ processes, and Markov processes on partitions related to the $z$-measures.

Détail

Publié le : 2015-01-01

MR 3300963 | Zbl 06412897

DOI : https://doi.org/10.1214/13-AIHP579
Classification:  60B20,  60G55

Bibliographie

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. U.S. Government Printing Office, Washington, DC, 1964. | MR 167642 | Zbl 0643.33001

[2] M. Adler, J. Delépine and P. Van Moerbeke. Dyson’s nonintersecting Brownian motions with a few outliers. Comm. Pure Appl. Math. 62 (2009) 334–395. | MR 2487852 | Zbl 1166.60048

[3] M. Adler, P. L. Ferrari and P. Van Moerbeke. Airy processes with wanderers and new universality classes. Ann. Probab. 38 (2010) 714–769. | MR 2642890 | Zbl 1200.60069

[4] M. Adler, P. L. Ferrari and P. Van Moerbeke. Non-intersecting random walks in the neighborhood of a symmetric tacnode. Ann. Probab. 41 (2013) 2599–2647. | MR 3112926 | Zbl 1279.60062

[5] M. Adler, K. Johansson and P. Van Moerbeke. Double Aztec diamonds and the tacnode process. Adv. Math. 252 (2014) 518–571. | MR 3144240 | Zbl 06284332

[6] G. W. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge, 2010. | MR 2760897 | Zbl 1184.15023

[7] A. Aptekarev, P. Bleher and A. Kuijlaars. Large $n$ limit of Gaussian random matrices with external source, part II. Comm. Math. Phys. 259 (2005) 367–389. | MR 2172687 | Zbl 1129.82014

[8] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 (2005) 1643–1697. | MR 2165575 | Zbl 1086.15022

[9] J. Baik, K. Liechty and G. Schehr. On the joint distribution of the maximum and its position of the Airy2 process minus a parabola. J. Math. Phys. 53 (2012) 083303. | MR 3012644 | Zbl 1278.82070

[10] A. Borodin. Determinantal point processes. In The Oxford Handbook of Random Matrix Theory. Oxford Univ. Press, London, 2011. | MR 2932631 | Zbl 1238.60055

[11] A. Borodin and M. Duits. Limits of determinantal processes near a tacnode. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 243–258. | Numdam | MR 2779404 | Zbl 1208.82039

[12] A. Borodin, P. L. Ferrari and M. Prähofer. Fluctuations in the discrete TASEP with periodic initial configurations and the $\operatorname{Airy} _{1}$ process. Int. Math. Res. Pap. 2007 (2007) rpm002. | MR 2334008

[13] A. Borodin, P. L. Ferrari, M. Prähofer and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (5–6) (2007) 1055–1080. | MR 2363389 | Zbl 1136.82028

[14] A. Borodin, P. L. Ferrari and T. Sasamoto. Transition between $\operatorname{Airy} _{1}$ and $\operatorname{Airy} _{2}$ processes and TASEP fluctuations. Comm. Pure Appl. Math. 61 (2008) 1603–1629. | MR 2444377 | Zbl 1214.82062

[15] A. Borodin and G. Olshanski. Point processes and the infinite symmetric group. Math. Res. Lett. 5 (1998) 799–816. | MR 1671191 | Zbl 1044.20501

[16] A. Borodin and G. Olshanski. Random partitions and the Gamma kernel. Adv. Math. 194 (1) (2005) 141–202. | MR 2141857 | Zbl 1128.60301

[17] A. Borodin and G. Olshanski. Stochastic dynamics related to Plancherel measures on partitions. In Representation Theory, Dynamical Systems, and Asymptotic Combinatorics. American Mathematical Society Translations—Series 2: Advances in the Mathematical Sciences 217 9–21. American Mathematical Society. Providence, 2006. | MR 2276098 | Zbl 1109.60041

[18] A. Borodin and G. Olshanski. Markov processes on partitions. Probab. Theory Related Fields 135 (2006) 84–152. | MR 2214152 | Zbl 1105.60052

[19] A. Borodin and S. Péché. Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys. 132 (2008) 275–290. | MR 2415103 | Zbl 1145.82021

[20] A. Borodin and E. M. Rains. Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 (2005) 291–317. | MR 2185331 | Zbl 1127.82017

[21] A. N. Borodin and P. Salminen. Handbook of Brownian Motion: Facts and Formulae, 2nd edition. Birkhäuser, Basel, 2002. | MR 1912205 | Zbl 0859.60001

[22] T. Bothner and K. Liechty. Tail decay for the distribution of the endpoint of a directed polymer. Nonlinearity 26 (2013) 1449–1472. | MR 3056134 | Zbl 1268.82045

[23] E. Brézin and S. Hikami. Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 57 (1998) 4140–4149. | MR 1618958

[24] E. Brézin and S. Hikami. Level spacing of random matrices in an external source. Phys. Rev. E 58 (1998) 7176–7185. | MR 1662382

[25] I. Corwin and A. Hammond. Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 (2014) 441–508. | MR 3152753 | Zbl 06261669

[26] I. Corwin, J. Quastel and D. Remenik. Continuum statistics of the $\operatorname{Airy} _{2}$ process. Comm. Math. Phys. 317 (2013) 347–362. | MR 3010187 | Zbl 1257.82112

[27] B. Eynard and M. L. Mehta. Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 31 (1998) 4449–4456. | MR 1628667 | Zbl 0938.15012

[28] P. L. Ferrari. The universal $\operatorname{Airy} _{1}$ and $\operatorname{Airy} _{2}$ processes in the totally asymmetric simple exclusion process. In Integrable Systems and Random Matrices: In Honor of Percy Deift. Contemporary Mathematics 458 321–332. American Mathematical Society, Providence, 2008. | MR 2411915 | Zbl 1145.82332

[29] P. L. Ferrari and B. Vető. Non-colliding Brownian bridges and the asymmetric tacnode process. Electron. J. Probab. 44 (2012) 44. | MR 2946151 | Zbl 1258.60011

[30] I. Gessel and G. Viennot. Binomial determinants, paths, and hook length formulae. Adv. Math. 58 (1985) 300–321. | MR 815360 | Zbl 0579.05004

[31] K. Johansson. Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 (2002) 225–280. | MR 1900323 | Zbl 1008.60019

[32] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277–329. | MR 2018275 | Zbl 1031.60084

[33] K. Johansson. Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier (Grenoble) 55 (2005) 2129–2145. | Numdam | MR 2187949 | Zbl 1083.60079

[34] K. Johansson. Non-colliding Brownian motions and the extended tacnode process. Comm. Math. Phys. 319 (2013) 231–267. | MR 3034030 | Zbl 1268.60104

[35] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus. Springer, Berlin, 1991. | MR 1121940 | Zbl 0734.60060

[36] S. Karlin and G. Mcgregor. Coincidence probabilities. Pacific J. Math. 9 (1959) 1141–1164. | MR 114248 | Zbl 0092.34503

[37] M. Katori and H. Tanemura. Noncolliding squared Bessel processes. J. Stat. Phys. 142 (2011) 592–615. | MR 2771046 | Zbl 1211.82036

[38] B. Lindström. On the vector representation of induced matroids. Bull. London Math. Soc. 5 (1973) 85–90. | MR 335313 | Zbl 0262.05018

[39] G. Moreno Flores, J. Quastel and D. Remenik. Endpoint distribution of directed polymers in 1$+$1 dimensions. Comm. Math. Phys. 317 (2013) 363–380. | MR 3010188 | Zbl 1257.82117

[40] T. Nagao and P. J. Forrester. Multilevel dynamical correlation function for Dyson’s Brownian motion model of random matrices. Phys. Lett. A 247 (1998) 42–46.

[41] A. Okounkov and N. Reshetikhin. Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 (2003) 581–603. | MR 1969205 | Zbl 1009.05134

[42] G. Olshanski. An introduction to harmonic analysis on the infinite symmetric group. In Asymptotic Combinatorics with Applications to Mathematical Physics. A. Vershik (Ed.). Lecture Notes in Math. 1815 127–160. Springer, Berlin, 2003. | MR 2009838 | Zbl 1035.05100

[43] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 (2002) 1071–1106. | MR 1933446 | Zbl 1025.82010

[44] S. Prolhac and H. Spohn. The one-dimensional KPZ equation and the Airy process. J. Stat. Mech. Theory Exp. 3 (2011) P03020. | MR 2801168

[45] J. Quastel and D. Remenik. Supremum of the $\operatorname{Airy} _{2}$ process minus a parabola on a half line. J. Stat. Phys. 150 (2013) 442–456. | MR 3024136 | Zbl 1263.82045

[46] J. Quastel and D. Remenik. Local behavior and hitting probabilities of the $\operatorname{Airy} _{1}$ process. Probab. Theory Related Fields 157 (2013) 605–634. | MR 3129799 | Zbl 1285.60095

[47] J. Quastel and D. Remenik. Tails of the endpoint distribution of directed polymers. Ann. Inst. Henri Poincaré Probab. Stat. To appear. | Numdam | MR 3300961 | Zbl 06412895

[48] J. Quastel and D. Remenik. Airy processes and variational problems. In Topics in Percolative and Disordered Systems. To appear. Available at arXiv:1301.0750. | MR 3229288

[49] T. Sasamoto. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 (2005) L549–L556. | MR 2165697

[50] G. Schehr. Extremes of $N$ vicious walkers for large $N$: Application to the directed polymer and KPZ interfaces. J. Stat. Phys. 149 (2012) 385–410. | MR 2992794 | Zbl 1259.82146

[51] G. Schehr, S. N. Majumdar, A. Comtet and J. Randon-Furling. Exact distribution of the maximal height of $p$ vicious walkers. Phys. Rev. Lett. 101 (2008) 150601. | MR 2460718 | Zbl 1228.82038

[52] B. Simon. Trace Ideals and Their Applications, 2nd edition. American Mathematical Society, Providence, 2000. | MR 2154153 | Zbl 1074.47001

[53] J. R. Stembridge. Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83 (1990) 96–131. | MR 1069389 | Zbl 0790.05007

[54] C. Tracy and H. Widom. Differential equations for the Dyson process. Comm. Math. Phys. 252 (2004) 7–41. | MR 2103903 | Zbl 1124.82007

[55] C. Tracy and H. Widom. The Pearcey process. Comm. Math. Phys. 263 (2006) 381–400. | MR 2207649 | Zbl 1129.82031