Exact solution of interacting particle systems related to random matrices. (English) Zbl 1521.60045
Summary: We consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular the motion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the space-inhomogeneous setting and for general initial condition this is the first time such a result has been proven. We moreover consider the model of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition.
References:
| [1] | Adler, M.; Nordenstam, E.; van Moerbeke, P., Consecutive minors for Dyson’s Brownian motions, Stoch. Process. Appl., 124, 6, 2023-2051 (2014) · Zbl 1308.60013 |
| [2] | Adler, M.; Nordenstam, E.; Van Moerbeke, P., The Dyson Brownian minor process, Ann. Inst. Fourier (Grenoble), 64, 3, 971-1009 (2014) · Zbl 1318.60008 |
| [3] | Adler, M.; van Moerbeke, P., Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum, Ann. Math. (2), 153, 1, 149-189 (2001) · Zbl 1033.82005 |
| [4] | Adler, M.; van Moerbeke, P., PDEs for the joint distributions of the Dyson, Airy and sine processes, Ann. Probab., 33, 4, 1326-1361 (2005) · Zbl 1093.60021 |
| [5] | Albanese, C.; Kuznetsov, A., Transformations of Markov processes and classification scheme for solvable driftless diffusions, Markov Process. Related Fields, 15, 4, 563-574 (2009) · Zbl 1195.60104 |
| [6] | Anderson, GW; Guionnet, A.; Zeitouni, O., An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1184.15023 |
| [7] | Arai, Y., The KPZ fixed point for discrete time TASEPs, J. Phys. A Math. Theor., 53, 41 (2020) · Zbl 1519.82064 |
| [8] | Arista, J.; Demni, N., Explicit expressions of the Hua-Pickrell semigroup, Theory Probab. Appl., 67, 2, 208-228 (2022) · Zbl 1513.76137 |
| [9] | Assiotis, T. A matrix Bougerol identity and the Hua-Pickrell measures. Electron. Commun. Probab. 23, Paper No. 7, 11 (2018) · Zbl 1398.60010 |
| [10] | Assiotis, T., Determinantal structures in space-inhomogeneous dynamics on interlacing arrays, Ann. Henri Poincaré, 21, 3, 909-940 (2020) · Zbl 1439.82033 |
| [11] | Assiotis, T., Hua-Pickrell diffusions and Feller processes on the boundary of the graph of spectra, Ann. Inst. Henri Poincaré Probab. Stat., 56, 2, 1251-1283 (2020) · Zbl 1434.60212 |
| [12] | Assiotis, T. On the singular values of complex matrix Brownian motion with a matrix drift. arXiv preprint arXiv:2107.05028 (to appear Bernoulli) (2021) · Zbl 1510.60072 |
| [13] | Assiotis, T., O’Connell, N., Warren, J. Interlacing diffusions. Sémin. Probab. L 301-380 (2019) · Zbl 1452.60049 |
| [14] | Avram, F.; Leonenko, NN; Šuvak, N., Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85, 2, 346-369 (2013) · Zbl 1291.60161 |
| [15] | Babusci, D., Dattoli, G., Górska, K., Penson, K.A.: Lacunary generating functions for the Laguerre polynomials. Sém. Lothar. Combin. 76, Art. B76b, 19 (2016-2019) · Zbl 1364.33013 |
| [16] | Bisi, E., Liao, Y., Saenz, A., Zygouras, N.: Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point. arXiv preprint arXiv:2208.13580 (2022) · Zbl 07719656 |
| [17] | Borodin, A.: Determinantal point processes, pp. 231-249. In: The Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011) · Zbl 1238.60055 |
| [18] | Borodin, A.; Deift, P., Fredholm determinants, Jimbo-Miwa-Ueno \(\tau \)-functions, and representation theory, Commun. Pure Appl. Math., 55, 9, 1160-1230 (2002) · Zbl 1033.34089 |
| [19] | Borodin, A.; Ferrari, PL, Large time asymptotics of growth models on space-like paths. I. PushASEP, Electron. J. Probab., 13, 50, 1380-1418 (2008) · Zbl 1187.82084 |
| [20] | Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the \({\rm Airy}_1\) process. Int. Math. Res. Pap. IMRP 1, Art. ID rpm002, 47 (2007) · Zbl 1136.82321 |
| [21] | Borodin, A.; Ferrari, PL; Prähofer, M.; Sasamoto, T., Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys., 129, 5-6, 1055-1080 (2007) · Zbl 1136.82028 |
| [22] | Borodin, A.; Ferrari, PL; Sasamoto, T., Large time asymptotics of growth models on space-like paths. II. PNG and parallel TASEP, Commun. Math. Phys., 283, 2, 417-449 (2008) · Zbl 1201.82030 |
| [23] | Borodin, A.; Olshanski, G., Infinite random matrices and ergodic measures, Commun. Math. Phys., 223, 1, 87-123 (2001) · Zbl 0987.60020 |
| [24] | Borodin, A.; Rains, EM, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys., 121, 3-4, 291-317 (2005) · Zbl 1127.82017 |
| [25] | Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae, 2nd edn. Probability and its Applications. Birkhäuser Verlag, Basel (2002) · Zbl 1012.60003 |
| [26] | Bougerol, P.: Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré Sect. B. (N.S.) 19(4), 369-391 (1983) · Zbl 0533.60010 |
| [27] | Bougerol, P.; Jeulin, T., Paths in Weyl chambers and random matrices, Probab. Theory Related Fields, 124, 4, 517-543 (2002) · Zbl 1020.15024 |
| [28] | Brézin, E.; Hikami, S., Extension of level-spacing universality, Phys. Rev. E, 56, 1, 264 (1997) |
| [29] | Bru, M-F, Wishart processes, J. Theor. Probab., 4, 4, 725-751 (1991) · Zbl 0737.60067 |
| [30] | Carmona, P.; Petit, F.; Yor, M., Beta-gamma random variables and intertwining relations between certain Markov processes, Rev. Mat. Iberoamericana, 14, 2, 311-367 (1998) · Zbl 0919.60074 |
| [31] | Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001, 76 (2012) · Zbl 1247.82040 |
| [32] | Dauvergne, D., Ortmann, J., Virág, B.: The directed landscape. arXiv preprint arXiv:1812.00309 (2018) · Zbl 1522.60075 |
| [33] | de Boor, C., Divided differences, Surv. Approx. Theory, 1, 46-69 (2005) · Zbl 1071.65027 |
| [34] | Demni, N., The Laguerre process and generalized Hartman-Watson law, Bernoulli, 13, 2, 556-580 (2007) · Zbl 1139.60037 |
| [35] | Dieker, AB; Warren, J., Determinantal transition kernels for some interacting particles on the line, Ann. Inst. Henri Poincaré Probab. Stat., 44, 6, 1162-1172 (2008) · Zbl 1181.60144 |
| [36] | Doob, J.L.: Classical potential theory and its probabilistic counterpart. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262. Springer, New York (1984) · Zbl 0549.31001 |
| [37] | Doumerc, Y.: Matrices aléatoires, processus stochastiques et groupes de réflexions. Ph.D. Thesis, Université Paul Sabatier-Toulouse III (2005) |
| [38] | Dufresne, D., The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J., 1-2, 39-79 (1990) · Zbl 0743.62101 |
| [39] | Dynkin, E.B.: Markov processes. Vols. I, II, vol. 122 of Die Grundlehren der mathematischen Wissenschaften, Band 121. Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone · Zbl 0132.37901 |
| [40] | Dyson, FJ, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys., 3, 1191-1198 (1962) · Zbl 0111.32703 |
| [41] | Erdős, L., Yau, H.-T.: A dynamical approach to random matrix theory, vol. 28 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2017) · Zbl 1379.15003 |
| [42] | Ethier, S.N., Kurtz, T.G.: Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986). Characterization and convergence · Zbl 0592.60049 |
| [43] | Ferrari, PL; Frings, R., On the partial connection between random matrices and interacting particle systems, J. Stat. Phys., 141, 4, 613-637 (2010) · Zbl 1205.82104 |
| [44] | Ferrari, PL; Spohn, H.; Weiss, T., Brownian motions with one-sided collisions: the stationary case, Electron. J. Probab., 20, 69, 41 (2015) · Zbl 1327.60188 |
| [45] | Ferrari, PL; Spohn, H.; Weiss, T., Scaling limit for Brownian motions with one-sided collisions, Ann. Appl. Probab., 25, 3, 1349-1382 (2015) · Zbl 1315.60108 |
| [46] | Forman, J.L., Sø rensen, M.: The Pearson diffusions: a class of statistically tractable diffusion processes. Scand. J. Stat. 35(3), 438-465 (2008) · Zbl 1198.62078 |
| [47] | Forrester, PJ, Log-Gases and Random Matrices. London Mathematical Society Monographs Series (2010), Princeton: Princeton University Press, Princeton · Zbl 1217.82003 |
| [48] | Forrester, P.J., Nagao, T.: Determinantal correlations for classical projection processes. J. Stat. Mech. Theory Exp. P08011 (2011) |
| [49] | Forrester, PJ; Nagao, T., Determinantal correlations for classical projection processes, J. Stat. Mech. Theory Exp., 2011, 8, P08011 (2011) |
| [50] | Forrester, PJ; Witte, NS, Application of the \(\tau \)-function theory of Painlevé equations to random matrices: \( \rm P_V, \rm P_{III}\), the LUE, JUE, and CUE, Commun. Pure Appl. Math., 55, 6, 679-727 (2002) · Zbl 1029.34087 |
| [51] | Forrester, PJ; Witte, NS, Application of the \(\tau \)-function theory of Painlevé equations to random matrices: \( {P}_{VI}\), the JUE, CyUE, cJUE and scaled limits, Nagoya Math. J., 174, 29-114 (2004) · Zbl 1056.15023 |
| [52] | Göing-Jaeschke, A.; Yor, M., A survey and some generalizations of Bessel processes, Bernoulli, 9, 2, 313-349 (2003) · Zbl 1038.60079 |
| [53] | Graczyk, P., Mał ecki, J. Multidimensional Yamada-Watanabe theorem and its applications to particle systems. J. Math. Phys. 54(2), 021503, 15 (2013) · Zbl 1296.35224 |
| [54] | Graczyk, P., Mał ecki, J. Strong solutions of non-colliding particle systems. Electron. J. Probab. 19(119), 21 (2014) · Zbl 1307.60079 |
| [55] | Gupta, A.K., Nagar, D.K.: Matrix variate distributions. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 104. Chapman & Hall/CRC, Boca Raton, FL (2000) · Zbl 0935.62064 |
| [56] | Hall, B.C., Ho, C.-W.: The heat flow conjecture for random matrices. arXiv preprint arXiv:2202.09660 (2022) |
| [57] | Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd edn., vol. 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1989) · Zbl 0684.60040 |
| [58] | Itô, K., McKean, Jr., H.P. Diffusion processes and their sample paths. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer, Berlin (1974). Second printing, corrected · Zbl 0285.60063 |
| [59] | Johansson, K., Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Commun. Math. Phys., 215, 3, 683-705 (2001) · Zbl 0978.15020 |
| [60] | Johansson, K., Non-intersecting, simple, symmetric random walks and the extended Hahn kernel, Ann. Inst. Fourier (Grenoble), 55, 6, 2129-2145 (2005) · Zbl 1083.60079 |
| [61] | Johansson, K.: Random matrices and determinantal processes. In: Mathematical Statistical Physics, pp. 1-55. Elsevier B.V., Amsterdam (2006) · Zbl 1411.60144 |
| [62] | Johansson, K.; Nordenstam, E., Eigenvalues of GUE minors, Electron. J. Probab., 11, 50, 1342-1371 (2006) · Zbl 1127.60047 |
| [63] | Johansson, K., Nordenstam, E.: Erratum to: “Eigenvalues of GUE minors” [Electron. J. Probab. 11 (2006), no. 50, 1342-1371; mr2268547]. Electron. J. Probab. 12, 1048-1051 (2007) · Zbl 1134.60344 |
| [64] | Kabluchko, Z.: Lee-yang zeroes of the curie-weiss ferromagnet, unitary hermite polynomials, and the backward heat flow. arXiv preprint arXiv:2203.05533 (2022) |
| [65] | Karlin, S.; McGregor, J., Coincidence probabilities, Pacific J. Math., 9, 1141-1164 (1959) · Zbl 0092.34503 |
| [66] | Karlin, S., Taylor, H.M.: A second course in stochastic processes. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1981) · Zbl 0469.60001 |
| [67] | Katori, M., Determinantal martingales and noncolliding diffusion processes, Stoch. Process. Appl., 124, 11, 3724-3768 (2014) · Zbl 1296.60213 |
| [68] | Katori, M., Elliptic determinantal process of type A, Probab. Theory Related Fields, 162, 3-4, 637-677 (2015) · Zbl 1416.60081 |
| [69] | Katori, M.; Tanemura, H., Non-equilibrium dynamics of Dyson’s model with an infinite number of particles, Commun. Math. Phys., 293, 2, 469-497 (2010) · Zbl 1214.82061 |
| [70] | Katori, M.; Tanemura, H., Noncolliding squared Bessel processes, J. Stat. Phys., 142, 3, 592-615 (2011) · Zbl 1211.82036 |
| [71] | Knizel, A.; Petrov, L.; Saenz, A., Generalizations of TASEP in discrete and continuous inhomogeneous space, Commun. Math. Phys., 372, 3, 797-864 (2019) · Zbl 1439.60093 |
| [72] | Kolmogoroff, A., Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104, 1, 415-458 (1931) · JFM 57.0613.03 |
| [73] | König, W.; O’Connell, N., Eigenvalues of the Laguerre process as non-colliding squared Bessel processes, Electron. Commun. Probab., 6, 107-114 (2001) · Zbl 1011.15012 |
| [74] | Marcus, A.W.: Finite free point processes. arXiv preprint arXiv:2205.00495 (2022) |
| [75] | Matetski, K., Quastel, J., Remenik, D.: Polynuclear growth and the Toda lattice. arXiv preprint arXiv:2209.02643 (2022) · Zbl 1505.82041 |
| [76] | Matetski, K., Remenik, D.: TASEP and generalizations: method for exact solution. Probab. Theory Related Fields 1-84 (2022) |
| [77] | Matetski, K.; Remenik, D.; Jeremy, Q., The KPZ fixed point, Acta Math., 227, 115-203 (2021) · Zbl 1505.82041 |
| [78] | Nica, M.; Quastel, J.; Remenik, D., One-sided reflected Brownian motions and the KPZ fixed point, Forum Math. Sigma, 8 (2020) · Zbl 1455.60131 |
| [79] | Nica, M.; Quastel, J.; Remenik, D., Solution of the Kolmogorov equation for TASEP, Ann. Probab., 48, 5, 2344-2358 (2020) · Zbl 1456.60263 |
| [80] | O’Connell, N.; Yor, M., Brownian analogues of Burke’s theorem, Stoch. Process. Appl., 96, 2, 285-304 (2001) · Zbl 1058.60078 |
| [81] | O’Connell, N.; Yor, M., A representation for non-colliding random walks, Electron. Commun. Probab., 7, 1-12 (2002) · Zbl 1037.15019 |
| [82] | Petrov, L.: PushTASEP in inhomogeneous space. Electron. J. Probab. 25, Paper No. 114, 25 (2020) · Zbl 1453.82058 |
| [83] | Pinsky, RG, Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0858.31001 |
| [84] | Quastel, J., Remenik, D.: Airy processes and variational problems. In: Topics in percolative and disordered systems, vol. 69 of Springer Proc. Math. Stat., pp. 121-171. Springer, New York (2014) · Zbl 1329.82059 |
| [85] | Quastel, J., Remenik, D.: KP governs random growth off a 1-dimensional substrate. Forum Math. Pi 10, Paper No. e10, 26 (2022) · Zbl 1502.60161 |
| [86] | Quastel, J., Sarkar, S.: Convergence of exclusion processes and the KPZ equation to the KPZ fixed point. J. Am. Math. Soc. (2022) · Zbl 1520.60063 |
| [87] | Revuz, D., Yor, M.: Continuous martingales and Brownian motion, 3rd edn., vol. 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999) · Zbl 0917.60006 |
| [88] | Rider, B.; Valkó, B., Matrix Dufresne identities, Int. Math. Res. Not. IMRN, 1, 174-218 (2016) · Zbl 1344.60070 |
| [89] | Rodgers, B., Tao, T.: The de Bruijn-Newman constant is non-negative. Forum Math. Pi 8, e6, 62 (2020) · Zbl 1454.11158 |
| [90] | Sasamoto, T., Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A, 38, 33, L549-L556 (2005) |
| [91] | Schütz, GM, Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys., 88, 1-2, 427-445 (1997) · Zbl 0945.82508 |
| [92] | Simon, B., Notes on infinite determinants of Hilbert space operators, Adv. Math., 24, 3, 244-273 (1977) · Zbl 0353.47008 |
| [93] | Spitzer, F., Interaction of Markov processes, Adv. Math., 5, 1970, 246-290 (1970) · Zbl 0312.60060 |
| [94] | Spohn, H.: Interacting., Brownian particles: a study of Dyson’s model. In: Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986): vol. 9 of IMA Vol. Math. Appl. Springer, New York, pp. 151-179 (1987) · Zbl 0674.60096 |
| [95] | Stroock, DW, Partial Differential Equations for Probabilists. Cambridge Studies in Advanced Mathematics (2008), Cambridge: Cambridge University Press, Cambridge · Zbl 1145.35002 |
| [96] | Tao, T.: Heat flow and zeroes of polynomials II: zeroes on a circle. blog post from june 2018, https://terrytao.wordpress.com/2018/06/07/ |
| [97] | Tracy, CA; Widom, H., Differential equations for Dyson processes, Commun. Math. Phys., 252, 1-3, 7-41 (2004) · Zbl 1124.82007 |
| [98] | Tsai, L-C, Infinite dimensional stochastic differential equations for Dyson’s model, Probab. Theory Related Fields, 166, 3-4, 801-850 (2016) · Zbl 1354.60064 |
| [99] | Tubikanec, I., Tamborrino, M., Lansky, P., Buckwar, E.: Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion. J. Comput. Appl. Math. 406, Paper No. 113951, 29 (2022) · Zbl 1490.60209 |
| [100] | Warren, J., Dyson’s Brownian motions, intertwining and interlacing, Electron. J. Probab., 12, 19, 573-590 (2007) · Zbl 1127.60078 |
| [101] | Weiss, T.; Ferrari, P.; Spohn, H., Reflected Brownian Motions in the KPZ Universality Class. SpringerBriefs in Mathematical Physics (2017), Cham: Springer, Cham · Zbl 1366.82004 |
| [102] | Witte, NS; Forrester, PJ, Gap probabilities in the finite and scaled Cauchy random matrix ensembles, Nonlinearity, 13, 6, 1965-1986 (2000) · Zbl 0980.15018 |
| [103] | Wong, E.: The construction of a class of stationary Markoff processes. In: Proceedings of Symposia in Applied Mathematics, vol. XVI, pp. 264-276. American Mathematical Society, Providence (1964) · Zbl 0139.34406 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
