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On the phase diagram of the polymer model

Arjun Krishnan, Sevak Mkrtchyan, Scott Neville

TL;DR

This paper studies how an external tilt affects the phase diagram of the directed polymer in iid weights, revealing tilt-induced transitions between Brownian (weak disorder) and localized (strong disorder) behavior at fixed temperature. It introduces the two-parameter Helmholtz free energy $g_{pl}(\beta,h)$ and proves its existence and convexity for general walks, including infinite-range cases. The authors show that for $d\ge 3$ there exist external fields $h$ for which both weak and strong disorder occur, and provide an entropy-based criterion $\mathcal{H}(\mathbb{Q}_\beta|\mathbb{P})>\mathcal{H}(q(h))$ guaranteeing strong disorder; for finite-range walks, one can choose fields with singleton maximizers to ensure localization, while Gaussian walks can behave differently depending on $\beta$. They also establish weak-disorder endpoint CLTs, a monotonicity property with respect to tilt, and present numerical experiments that illustrate the tilt-driven phase transitions and the resulting phase diagram. Overall, the work uncovers a rich, tilt-driven phase structure in directed polymers and highlights the possibility of driving phase transitions through an external field, with implications for polymer behavior in heterogeneous media.

Abstract

In dimensions 3 or larger, it is a classical fact that the directed polymer model has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can induce the phase transition in the directed polymer model.

On the phase diagram of the polymer model

TL;DR

This paper studies how an external tilt affects the phase diagram of the directed polymer in iid weights, revealing tilt-induced transitions between Brownian (weak disorder) and localized (strong disorder) behavior at fixed temperature. It introduces the two-parameter Helmholtz free energy and proves its existence and convexity for general walks, including infinite-range cases. The authors show that for there exist external fields for which both weak and strong disorder occur, and provide an entropy-based criterion guaranteeing strong disorder; for finite-range walks, one can choose fields with singleton maximizers to ensure localization, while Gaussian walks can behave differently depending on . They also establish weak-disorder endpoint CLTs, a monotonicity property with respect to tilt, and present numerical experiments that illustrate the tilt-driven phase transitions and the resulting phase diagram. Overall, the work uncovers a rich, tilt-driven phase structure in directed polymers and highlights the possibility of driving phase transitions through an external field, with implications for polymer behavior in heterogeneous media.

Abstract

In dimensions 3 or larger, it is a classical fact that the directed polymer model has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can induce the phase transition in the directed polymer model.
Paper Structure (14 sections, 17 theorems, 108 equations, 2 figures, 1 table)

This paper contains 14 sections, 17 theorems, 108 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical results
  3. The external field in the polymer model
  4. Gibbs and Helmholtz Free Energies
  5. The effect of the external field
  6. Main Results
  7. Acknowledgements
  8. Existence of the free-energy
  9. Weak disorder
  10. Endpoint behavior in weak disorder
  11. Strong disorder
  12. Gaussian random walk
  13. Monotonicity
  14. Numerical Results

Key Result

Theorem 1.1

Let where $S_n$ and $S_n'$ are independent copies of the random walk with transition kernel $p$. If $\pi(p) < 1$ and then $W_\infty > 0$ a.s.

Figures (2)

  • Figure 1: This is a schematic of the weak and strong disorder regions as a function of the external field $h\in\mathbb{R}^3$ on the plane $h_3=0$ when $d=3$, $p$ is the simple random walk and $\beta$ is small enough to be in the $L^2$ region. $D$ is contained in the diagonal lines $h_1 = \pm h_2$. As long as $\beta$ is small enough, there is region of weak disorder for small $|h|$. If $h$ is not in $D$, our results say that $\lambda h$ is at low-temperature and hence in $\operatorname{STRONG}_\beta$ for $\lambda$ large enough. Our results cannot precisely pin down the phase transition in $h$ with $\beta$ fixed; and so in the region between $\operatorname{WEAK}_\beta$ and $\operatorname{STRONG}_\beta$ is left unshaded and unpatterned.
  • Figure 2: $g_{pl}(\beta,h)$ is in blue (solid), and the RHS of \ref{['eq:annealed bound for gpl']} is in green (dotted), each as a function of $h = t e_1$ From left to right, $\beta$ takes the values $1,3,\textrm{ and } 5$.

Theorems & Definitions (36)

  • Theorem 1.1: MR1006293MR1413246
  • Theorem 1.2: MR968950MR1006293MR1413246MR2271480MR1371075
  • Proposition 1.3
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 2.6
  • Theorem 2.7
  • ...and 26 more