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Crystallization of discrete $N$-particle systems at high temperature

Cesar Cuenca, Maciej Dołęga

Abstract

This is the second paper in a series studying the global asymptotics of discrete $N$-particle systems with inverse temperature parameter $θ$ in the high temperature regime. In the first paper, we established necessary and sufficient conditions for the Law of Large Numbers at high temperature in terms of Jack generating functions. In this paper, we derive a functional equation for the moment generating function of the limiting measure, which enables its analysis using analytic tools. We apply this functional equation to compute the densities of the high temperature limits of the pure Jack measures. As a special case, we obtain the high temperature limit of the large fixed-time distribution of the discrete-space $β$-Dyson Brownian motion of Gorin-Shkolnikov. Two special cases of our densities are the high temperature limits of discrete versions of the G$β$E, computed by Allez-Bouchaud-Guionnet in [Phys. Rev. Lett. 109 (2012), 094102; arXiv:1205.3598], and L$β$E, computed by Allez-Bouchaud-Majumdar-Vivo in [J. Phys. A, vol. 46, no. 1 (2013), 015001; arXiv:1209.6171]. Moreover, we prove the following crystallization phenomenon of the particles in the high temperature limit: the limiting measures are uniformly supported on disjoint intervals with unit gaps and their locations correspond to the zeros of explicit special functions with all roots located in the real line. We also show that these zeros correspond to the spectra of certain unbounded Jacobi operators.

Crystallization of discrete $N$-particle systems at high temperature

Abstract

This is the second paper in a series studying the global asymptotics of discrete -particle systems with inverse temperature parameter in the high temperature regime. In the first paper, we established necessary and sufficient conditions for the Law of Large Numbers at high temperature in terms of Jack generating functions. In this paper, we derive a functional equation for the moment generating function of the limiting measure, which enables its analysis using analytic tools. We apply this functional equation to compute the densities of the high temperature limits of the pure Jack measures. As a special case, we obtain the high temperature limit of the large fixed-time distribution of the discrete-space -Dyson Brownian motion of Gorin-Shkolnikov. Two special cases of our densities are the high temperature limits of discrete versions of the GE, computed by Allez-Bouchaud-Guionnet in [Phys. Rev. Lett. 109 (2012), 094102; arXiv:1205.3598], and LE, computed by Allez-Bouchaud-Majumdar-Vivo in [J. Phys. A, vol. 46, no. 1 (2013), 015001; arXiv:1209.6171]. Moreover, we prove the following crystallization phenomenon of the particles in the high temperature limit: the limiting measures are uniformly supported on disjoint intervals with unit gaps and their locations correspond to the zeros of explicit special functions with all roots located in the real line. We also show that these zeros correspond to the spectra of certain unbounded Jacobi operators.

Paper Structure

This paper contains 26 sections, 25 theorems, 172 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. LLN for discrete N-particle systems at high temperature and the gamma-quantized R-transform
  4. Crystallization phenomenon for non-intersecting random walks at high temperature
  5. Pure Jack measures
  6. Jack polynomials and Jack generating functions
  7. Partitions
  8. Algebra of symmetric functions
  9. Jack symmetric functions
  10. Jack generating functions
  11. Jack-positive specializations
  12. Law of Large Numbers at high temperature for discrete N-particle ensembles
  13. The statement of the LLN
  14. Application to pure Jack measures
  15. Pure alpha Jack measures
  16. ...and 11 more sections

Key Result

Theorem 1

Assume that the sequences $(m_n)_{n\ge 1}$ and $(\kappa^\gamma_n)_{n\ge 1}$ satisfy the relations eq:intro_moms_cums. Then there exists a unique sequence $(c^\gamma_n)_{n\ge 1}$ such that we have the following equalities of formal power series in $z$ and $z^{-1}$, respectively, where $z^{\uparrow n} := \prod_{i=0}^{n-1}{(z+i)}$ and $\widetilde{\mathcal{L}}\left\{ \sum_{n\ge 0}{\frac{a_n}{n!} t^n}

Figures (5)

  • Figure 1: Three simulations of the Gorin--Shkolnikov process with $N=60$ particles and trivial initial configuration. The simulations show the asymptotic behavior in the fixed temperature regime on the left with $\theta=1$, and in the high temperature regime with $\gamma = 2$ in the middle, and $\gamma = \frac{1}{2}$ on the right. The $x$-axis represents the space and the $y$-axis represents the time.
  • Figure 2: A simulation of the Gorin--Shkolnikov process with $N=300$ particles and trivial initial configuration $\mathscr{L}^{(0)}_i = \theta(1-i)$, for $1 \leq i \leq N$, in the high temperature regime with $\theta = \frac{2}{N}$. Note that the initial configuration converges to the uniform distribution on $[-2,0]$. On the right-hand side, we see the distributions of particles at times $t\approx\frac{\eta}{\theta}$, for $\eta = 1/2$ and $\eta = 1$, which seem to be supported on the union of disjoint intervals with gaps between consecutive ones having length close to $1$.
  • Figure 3: Juxtaposition of the simulation (in blue) from \ref{['fig:Markov_Simulation']} at time $t\approx\frac{\eta}{\theta}$ and the graph (in green) of the confluent hypergeometric function ${}_1F_1(\gamma;z;-\eta)$, where $\eta = 1/2$, in the real variable $z$. \ref{['thm:application_plancherel']} predicts that the right endpoints of the intervals on the picture approximate the zeroes of ${}_1F_1(\gamma;z;-\eta)$, which are all real.
  • Figure 4: The same comparison as in \ref{['fig:Comparison1/2']} for $\eta = 1$.
  • Figure 5: The five Łukasiewicz paths of length $3$, the weights associated to each of their steps, and the overall contribution after applying the divided difference operator $\Delta_\gamma$. The horizontal steps at height $0$ and their weight after applying the divided difference operator are marked in blue.

Theorems & Definitions (56)

  • Theorem 1: \ref{['thm:r_transform']} in the text
  • Definition 1
  • Definition 2
  • Example 1
  • Example 2
  • Example 3
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 4
  • ...and 46 more