Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Multiplicative functionals on ensembles of non-intersecting paths

Alexei Borodin, Ivan Corwin, and Daniel Remenik

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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 $\operatorname{Airy}_{2\to1}$ processes, and Markov processes on partitions related to the $z$-measures.


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 $\operatorname{Airy}_{2\to1}$ ainsi que les processus de Markov sur les partitions reliées aux $z$-mesures.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 1 (2015), 28-58.

First available in Project Euclid: 14 January 2015

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60G55: Point processes

Non-intersecting paths Determinantal point process


Borodin, Alexei; Corwin, Ivan; Remenik, Daniel. Multiplicative functionals on ensembles of non-intersecting paths. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 28--58. doi:10.1214/13-AIHP579.

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  • [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.
  • [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.
  • [3] M. Adler, P. L. Ferrari and P. van Moerbeke. Airy processes with wanderers and new universality classes. Ann. Probab. 38 (2010) 714–769.
  • [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.
  • [5] M. Adler, K. Johansson and P. van Moerbeke. Double Aztec diamonds and the tacnode process. Adv. Math. 252 (2014) 518–571.
  • [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.
  • [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.
  • [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.
  • [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.
  • [10] A. Borodin. Determinantal point processes. In The Oxford Handbook of Random Matrix Theory. Oxford Univ. Press, London, 2011.
  • [11] A. Borodin and M. Duits. Limits of determinantal processes near a tacnode. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 243–258.
  • [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.
  • [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.
  • [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.
  • [15] A. Borodin and G. Olshanski. Point processes and the infinite symmetric group. Math. Res. Lett. 5 (1998) 799–816.
  • [16] A. Borodin and G. Olshanski. Random partitions and the Gamma kernel. Adv. Math. 194 (1) (2005) 141–202.
  • [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.
  • [18] A. Borodin and G. Olshanski. Markov processes on partitions. Probab. Theory Related Fields 135 (2006) 84–152.
  • [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.
  • [20] A. Borodin and E. M. Rains. Eynard–Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 (2005) 291–317.
  • [21] A. N. Borodin and P. Salminen. Handbook of Brownian Motion: Facts and Formulae, 2nd edition. Birkhäuser, Basel, 2002.
  • [22] T. Bothner and K. Liechty. Tail decay for the distribution of the endpoint of a directed polymer. Nonlinearity 26 (2013) 1449–1472.
  • [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.
  • [24] E. Brézin and S. Hikami. Level spacing of random matrices in an external source. Phys. Rev. E 58 (1998) 7176–7185.
  • [25] I. Corwin and A. Hammond. Brownian Gibbs property for Airy line ensembles. Invent. Math. 195 (2014) 441–508.
  • [26] I. Corwin, J. Quastel and D. Remenik. Continuum statistics of the $\operatorname{Airy} _{2}$ process. Comm. Math. Phys. 317 (2013) 347–362.
  • [27] B. Eynard and M. L. Mehta. Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 31 (1998) 4449–4456.
  • [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.
  • [29] P. L. Ferrari and B. Vető. Non-colliding Brownian bridges and the asymmetric tacnode process. Electron. J. Probab. 44 (2012) 44.
  • [30] I. Gessel and G. Viennot. Binomial determinants, paths, and hook length formulae. Adv. Math. 58 (1985) 300–321.
  • [31] K. Johansson. Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 (2002) 225–280.
  • [32] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277–329.
  • [33] K. Johansson. Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier (Grenoble) 55 (2005) 2129–2145.
  • [34] K. Johansson. Non-colliding Brownian motions and the extended tacnode process. Comm. Math. Phys. 319 (2013) 231–267.
  • [35] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus. Springer, Berlin, 1991.
  • [36] S. Karlin and G. McGregor. Coincidence probabilities. Pacific J. Math. 9 (1959) 1141–1164.
  • [37] M. Katori and H. Tanemura. Noncolliding squared Bessel processes. J. Stat. Phys. 142 (2011) 592–615.
  • [38] B. Lindström. On the vector representation of induced matroids. Bull. London Math. Soc. 5 (1973) 85–90.
  • [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.
  • [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.
  • [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.
  • [43] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 (2002) 1071–1106.
  • [44] S. Prolhac and H. Spohn. The one-dimensional KPZ equation and the Airy process. J. Stat. Mech. Theory Exp. 3 (2011) P03020.
  • [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.
  • [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.
  • [47] J. Quastel and D. Remenik. Tails of the endpoint distribution of directed polymers. Ann. Inst. Henri Poincaré Probab. Stat. To appear.
  • [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.
  • [49] T. Sasamoto. Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38 (2005) L549–L556.
  • [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.
  • [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.
  • [52] B. Simon. Trace Ideals and Their Applications, 2nd edition. American Mathematical Society, Providence, 2000.
  • [53] J. R. Stembridge. Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83 (1990) 96–131.
  • [54] C. Tracy and H. Widom. Differential equations for the Dyson process. Comm. Math. Phys. 252 (2004) 7–41.
  • [55] C. Tracy and H. Widom. The Pearcey process. Comm. Math. Phys. 263 (2006) 381–400.