Metadata
Abstract
The backward heat flow on the real line started from the initial condition $z^n$ results in the classical $n^{\rm th}$ Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large $n$ limit. Similarly, the backward heat flow with the periodic initial condition $(\sin \frac{\theta }{2})^n$ leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie–Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the $n^{\rm th}$ unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group $U(n)$. We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie–Weiss model in a complex external field. We identify the global distribution of the Lee–Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).
References
[ABH04] Real zero polynomials and Pólya–Schur type theorems, J. Anal. Math., Volume 94 (2004), pp. 49-60 | DOI | MR | Zbl
[AFU24] New combinatorial identity for the set of partitions and limit theorems in finite free probability theory, Int. Math. Res. Not., Volume 2024 (2024) no. 14, pp. 10524-10558 | DOI | MR | Zbl
[AGN21] Characteristic polynomials of products of non-Hermitian Wigner matrices: finite- results and Lyapunov universality, Electron. Commun. Probab., Volume 26 (2021), 30 | DOI | MR | Zbl
[AGVP23] Finite free cumulants: multiplicative convolutions, genus expansion and infinitesimal distributions, Trans. Am. Math. Soc., Volume 376 (2023) no. 6, pp. 4383-4420 | DOI | MR | Zbl
[AGZ10] An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, 2010 | MR | Zbl
[AKM12] Interacting particles on the line and Dunkl intertwining operator of type A: application to the freezing regime, J. Phys. A. Math. Theor., Volume 45 (2012) no. 39, 395201 | DOI | MR | Zbl
[Aom87] Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal., Volume 18 (1987) no. 2, pp. 545-549 | DOI | MR | Zbl
[AP18] Cumulants for finite free convolution, J. Comb. Theory, Ser. A, Volume 155 (2018), pp. 244-266 | DOI | MR | Zbl
[Ben34] Über lineare, verschiebungstreue Funktionaloperationen und die Nullstellen ganzer Funktionen, Comment. Math. Helv., Volume 7 (1934) no. 1, pp. 243-289 | DOI | MR | Zbl
[BH00] Characteristic polynomials of random matrices, Commun. Math. Phys., Volume 214 (2000) no. 1, pp. 111-135 | DOI | MR | Zbl
[BHS24] Rodrigues’ descendants of a polynomial and Boutroux curves, Constr. Approx., Volume 59 (2024) no. 3, pp. 737-798 | DOI | MR | Zbl
[Bia97a] Free Brownian motion, free stochastic calculus and random matrices, Free probability theory (Waterloo, ON, 1995) (Fields Institute Communications), Volume 12, American Mathematical Society, 1997, pp. 1-19 | MR | Zbl
[Bia97b] Segal–Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems, J. Funct. Anal., Volume 144 (1997) no. 1, pp. 232-286 | DOI | MR | Zbl
[Bia09] Matrix valued Brownian motion and a paper by Pólya, Séminaire de Probabilités XLII (Lecture Notes in Mathematics), Volume 1979, Springer, 2009, pp. 171-185 | DOI | Zbl
[Bil68] Convergence of probability measures, John Wiley & Sons, 1968 | MR | Zbl
[BLR17] Lee–Yang zeros for the DHL and 2D rational dynamics. I. Foliation of the physical cylinder, J. Math. Pures Appl. (9), Volume 107 (2017) no. 5, pp. 491-590 | DOI | MR | Zbl
[BLR20] Lee–Yang–Fisher zeros for the DHL and 2D rational dynamic. II. Global pluripotential interpretation, J. Geom. Anal., Volume 30 (2020) no. 1, pp. 777-833 | DOI | MR | Zbl
[Bui24] Efficient circular Dyson Brownian motion algorithm, Phys. Rev. Res., Volume 6 (2024) no. 2, 023264 | DOI
[Bur79] An introduction to classical complex analysis. Vol. 1, Pure and Applied Mathematics, 82, Academic Press Inc., 1979 | DOI | MR | Zbl
[BV92] Lévy–Hinčin type theorems for multiplicative and additive free convolution, Pac. J. Math., Volume 153 (1992) no. 2, pp. 217-248 | DOI | MR | Zbl
[CDG01] Large deviations upper bounds for the laws of matrix-valued processes and non-communicative entropies, Ann. Probab., Volume 29 (2001) no. 3, pp. 1205-1261 | DOI | MR | Zbl
[CHJR19] Limiting measure of Lee–Yang zeros for the Cayley tree, Commun. Math. Phys., Volume 370 (2019) no. 3, pp. 925-957 | DOI | MR | Zbl
[CL01] Brownian particles with electrostatic repulsion on the circle: Dyson’s model for unitary random matrices revisited, ESAIM, Probab. Stat., Volume 5 (2001), pp. 203-224 | DOI | Numdam | MR | Zbl
[Céb16] Matricial model for the free multiplicative convolution, Ann. Probab., Volume 44 (2016) no. 4, pp. 2427-2478 | DOI | MR | Zbl
[DBF20] Lee–Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model, Phys. Rev. B, Volume 102 (2020) no. 17, 174418 | DOI
[DE02] Matrix models for beta ensembles, J. Math. Phys., Volume 43 (2002) no. 11, pp. 5830-5847 | DOI | MR | Zbl
[DE05] Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 41 (2005) no. 6, pp. 1083-1099 | DOI | Numdam | MR | Zbl
[DF20] Lee–Yang theory of the Curie–Weiss model and its rare fluctuations, Phys. Rev. Res., Volume 2 (2020) no. 3, 033009 | DOI
[DG04] Random matrices, magic squares and matching polynomials, Electron. J. Comb., Volume 11 (2004) no. 2, 2 | MR | Zbl
[DM24] Statistical Mechanics of Mean-Field Disordered Systems: A Hamilton–Jacobi Approach, EMS Zurich Lectures in Advanced Mathematics Series, European Mathematical Society, 2024 | DOI | MR | Zbl
[DMTW19] Complex Burgers equation: A probabilistic perspective, Sojourns in Probability Theory and Statistical Physics. I. Spin Glasses and Statistical Mechanics, A Festschrift for Charles M. Newman (Springer Proceedings in Mathematics & Statistics), Volume 298, Springer (2019), pp. 138-170 | DOI | Zbl
[Dys62] A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys., Volume 3 (1962), pp. 1191-1198 | DOI | MR | Zbl
[Ede89] Eigenvalues and condition numbers of random matrices, Ph. D. Thesis, Massachusetts Institute of Technology, Cambridge, USA (1989) https://dspace.mit.edu/handle/1721.1/14322
[Ell06] Entropy, large deviations, and statistical mechanics, Classics in Mathematics, Springer, 2006 (reprint of the 1985 original) | DOI | MR | Zbl
[EN78a] Limit theorems for sums of dependent random variables occurring in statistical mechanics, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 44 (1978) no. 2, pp. 117-139 | DOI | MR | Zbl
[EN78b] The statistics of Curie–Weiss models, J. Stat. Phys., Volume 19 (1978) no. 2, pp. 149-161 | DOI | MR
[ENR80] Limit theorems for sums of dependent random variables occurring in statistical mechanics. II. Conditioning, multiple phases, and metastability, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 51 (1980) no. 2, pp. 153-169 | DOI | MR | Zbl
[FG06] Counting formulas associated with some random matrix averages, J. Comb. Theory, Ser. A, Volume 113 (2006) no. 6, pp. 934-951 | DOI | MR | Zbl
[FHS20] Monotone increment processes, classical Markov processes, and Loewner chains, Diss. Math., Volume 552 (2020), pp. 1-119 | DOI | MR | Zbl
[Fis65] The nature of critical points, Statistical Physics, Weak interactions, Field Theory (Brittin, W. E., ed.) (Lecture Notes in Theoretical Physics), Volume 7C, University of Colorado Press, 1965, pp. 1-159
[FKLZ24] Dip-ramp-plateau for Dyson Brownian motion from the identity on , Probab. Math. Phys., Volume 5 (2024) no. 2, pp. 321-355 | DOI | MR | Zbl
[FV18] Statistical mechanics of lattice systems. A concrete mathematical introduction, Cambridge University Press, 2018 | MR | Zbl
[Gal22] Modeling complex root motion of real random polynomials under differentiation, ISSAC’22 — Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, ACM Press (2022), pp. 245-253 | DOI | MR | Zbl
[Gaw87] On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials, J. Approx. Theory, Volume 50 (1987) no. 3, pp. 214-231 | DOI | MR | Zbl
[GK24] Universal objects of the infinite beta random matrix theory, J. Eur. Math. Soc., Volume 26 (2024) no. 9, pp. 3429-3496 | DOI | MR | Zbl
[GPS86] Complex temperature plane zeros in the mean-field approximation, J. Stat. Phys., Volume 45 (1986) no. 3-4, pp. 451-457 | DOI | MR
[Hal18] The eigenvalue process for Brownian motion in , Preprint at https://www3.nd.edu/~bhall/publications, 2018 | Zbl
[HH22] The heat flow conjecture for random matrices (2022) | arXiv
[HJN23] Motion of Lee–Yang Zeros, J. Stat. Phys., Volume 190 (2023) no. 3, 56 | MR | Zbl
[HK23] Dynamics of zeroes under repeated differentiation, Exp. Math., Volume 32 (2023) no. 4, pp. 573-599 | DOI | MR | Zbl
[HW96] Non-colliding Brownian motions on the circle, Bull. Lond. Math. Soc., Volume 28 (1996) no. 6, pp. 643-650 | DOI | MR | Zbl
[Kab21] Repeated differentiation and free unitary Poisson process (2021) | arXiv
[KBHK16] Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks, J. Phys. A. Math. Theor., Volume 49 (2016) no. 13, 135001 | DOI | MR | Zbl
[Kem17] Heat kernel empirical laws on and , J. Theor. Probab., Volume 30 (2017) no. 2, pp. 397-451 | DOI | MR | Zbl
[KM16] Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law, J. Approx. Theory, Volume 211 (2016), pp. 29-41 | DOI | MR | Zbl
[KT20] The Flow of Polynomial Roots Under Differentiation (2020) | arXiv
[LY52] Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev., II. Ser., Volume 87 (1952), pp. 410-419 | DOI | MR | Zbl
[Lév08] Schur–Weyl duality and the heat kernel measure on the unitary group, Adv. Math., Volume 218 (2008) no. 2, pp. 537-575 | DOI | MR | Zbl
[Mar18] Polynomial convolutions and (finite) free probability (2018) | arXiv
[Mir21] Hermitian, Non-Hermitian and Multivariate Finite Free Probability, Ph. D. Thesis, Princeton University, Princeton, USA (2021) https://dataspace.princeton.edu/...
[MN15] Mod-Gaussian convergence and its applications for models of statistical mechanics, In memoriam Marc Yor — Séminaire de Probabilités XLVII (Lecture Notes in Mathematics), Volume 2137, Springer, 2015, pp. 369-425 | DOI | Zbl
[MSS22] Finite free convolutions of polynomials, Probab. Theory Relat. Fields, Volume 182 (2022) no. 3-4, p. 807–848 | DOI | MR | Zbl
[New86] Shock waves and mean field bounds: Concavity and analyticity of the magnetization at low temperature. Appendix to Percolation theory: A selective survey of rigorous results, Advances in multiphase flow and related problems (Papanicolauo, George C., ed.), Society for Industrial and Applied Mathematics (1986), pp. 147-167 | Zbl
[NS06] Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, 2006 | DOI | MR | Zbl
[OS21] A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation, Proc. Am. Math. Soc., Volume 149 (2021), pp. 1581-1592 | DOI | Zbl
[PB21] A first course in random matrix theory: for physicists, engineers and data scientists, Cambridge University Press, 2021 | DOI | Zbl
[Rue69] Statistical mechanics. Rigorous results, The Mathematical Physics Monographs Series, W. A. Benjamin, Inc., 1969 | MR | Zbl
[RY99] Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, 293, Springer, 1999 | DOI | MR | Zbl
[RŚ15] Jeu de taquin dynamics on infinite Young tableaux and second class particles, Ann. Probab., Volume 43 (2015) no. 2, pp. 682-737 | DOI | MR | Zbl
[SFS85] Lectures on the theory of functions of a complex variable, Mir Publishers, 1985 | MR
[Ste19] A nonlocal transport equation describing roots of polynomials under differentiation, Proc. Am. Math. Soc., Volume 147 (2019) no. 11, pp. 4733-4744 | DOI | MR | Zbl
[Ste20] Conservation Laws for the Density of Roots of Polynomials under Differentiation (2020) | arXiv
[Ste23] Free convolution powers via roots of polynomials, Exp. Math., Volume 32 (2023) no. 4, pp. 567-572 | DOI | MR | Zbl
[SZ18] The Curie–Weiss model with complex temperature: phase transitions, J. Stat. Phys., Volume 172 (2018) no. 2, pp. 569-591 | DOI | MR | Zbl
[Sze75] Orthogonal polynomials, Colloquium Publications, 23, American Mathematical Society, 1975 | MR | Zbl
[Tao17] Heat flow and zeroes of polynomials, 2017 https://terrytao.wordpress.com/...
[Tao18] Heat flow and zeroes of polynomials II: zeroes on a circle, 2018 (heat-flow-and-zeroes-of-polynomials-ii-zeroes-on-a-circle/)
[Ull80] Orthogonal polynomials associated with an infinite interval, Mich. Math. J., Volume 27 (1980) no. 3, pp. 353-363 | DOI | MR | Zbl
[VDN92] Free random variables, CRM Monograph Series, 1, American Mathematical Society, 1992 | DOI | MR | Zbl
[Voi85] Symmetries of some reduced free product -algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) (Lecture Notes in Mathematics), Volume 1132, Springer, 1985, pp. 556-588 | DOI | MR | Zbl
[Voi87] Multiplication of certain noncommuting random variables, J. Oper. Theory, Volume 18 (1987) no. 2, pp. 223-235 | MR | Zbl
[VW20] Functional central limit theorems for multivariate Bessel processes in the freezing regime, Stochastic Anal. Appl., Volume 39 (2020) no. 1, pp. 136-156 | DOI | MR | Zbl
[VW22] Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions, Stochastic Processes Appl., Volume 143 (2022), pp. 207-253 | DOI | MR | Zbl
[YL52] Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev., II. Ser., Volume 87 (1952), pp. 404-409 | DOI | MR | Zbl
[Zho14] Free Brownian motion and free convolution semigroups: multiplicative case, Pac. J. Math., Volume 269 (2014) no. 1, pp. 219-256 | DOI | MR | Zbl
[Zho15] On the free convolution with a free multiplicative analogue of the normal distribution, J. Theor. Probab., Volume 28 (2015) no. 4, pp. 1354-1379 | DOI | MR | Zbl