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Large deviations for the log-Gamma polymer

Tom Claeys, Julian Mauersberger

Abstract

We conjecture an explicit expression for the lower tail large deviation rate function of the partition function of the log-Gamma polymer. We rigorously prove our result, except for one step for which we only provide heuristic evidence. We show that the large deviation rate function matches with that of last passage percolation with exponential weights in the zero-temperature limit, and with the lower tail of the Tracy-Widom distribution for moderate deviations.

Large deviations for the log-Gamma polymer

Abstract

We conjecture an explicit expression for the lower tail large deviation rate function of the partition function of the log-Gamma polymer. We rigorously prove our result, except for one step for which we only provide heuristic evidence. We show that the large deviation rate function matches with that of last passage percolation with exponential weights in the zero-temperature limit, and with the lower tail of the Tracy-Widom distribution for moderate deviations.

Paper Structure

This paper contains 29 sections, 15 theorems, 191 equations, 7 figures.

Table of Contents

  1. Introduction and main results
  2. Definition of the model.
  3. Known asymptotic results.
  4. Main result.
  5. Methodology.
  6. RH characterization of the step function Fredholm determinant
  7. Steepest descent analysis
  8. Transformation $Y \mapsto U$
  9. Transformation $U \mapsto T$
  10. Construction of the $g$-function for $0<s<-\theta\psi(\theta)$
  11. Transformation $T\mapsto S$
  12. Construction of parametrices for $\epsilon\leq s< -\theta\psi(\theta)-\epsilon$
  13. The global parametrix
  14. Local Airy parametrices
  15. The transformation $S \mapsto R$
  16. ...and 14 more sections

Key Result

Theorem 1.6

There exists $\theta_0>0$ such that as $n \to +\infty$ uniformly for $\epsilon <s< -\theta\psi(\theta)$ and $0<\theta<\theta_0$, for any $\epsilon>0$. The function $F(s,\theta)$ is given by eq: def f--def:F with $b(s,\theta)>0$ the unique positive number solving the equation eq: defining equation for b. Furthermore, the function $F(s,

Figures (7)

  • Figure 1: The log-Gamma polymer lattice for $n=6$, with a possible up-right path.
  • Figure 2: The shape of the jump contour $(-i\delta+\mathbb R)\cup\gamma$ for the RH problem for $U$.
  • Figure 3: Plot of regions in the complex $\zeta$-plane where $\Re h(\zeta;s,0)$ is positive (shaded) and negative (white). The left figure corresponds to $s>1$, the middle figure to $s=1$, and the right figure to $s<1$. $\mathcal{H}_+^s$ is the shaded region in the upper half plane, and $\mathcal{H}_-^s$ is the white region in the lower half plane.
  • Figure 4: The shape of the jump contour $\Sigma=\Sigma^+\cup\Sigma^0\cup\Sigma^-$ for the RH problem for $T$, in the cases $s>-\theta\psi(\theta)$ (left), $-s= -\theta\psi(\theta)$ (middle), and $0<s<-\theta\psi(\theta)$ (right).
  • Figure 5: Jump contour $\Sigma_R$ for $R$.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Conjecture 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1: Differential identity
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 20 more