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On the duality between particles and polymers

Mustazee Rahman

TL;DR

The paper establishes a comprehensive duality between tasep-like particle systems and last passage percolation (LPP) polymers across Geometric, Exponential, and Brownian settings, using a variational (Hopf–Lax–Oleinik–style) framework. It develops a language and topology for LPP models, builds isomorphisms and space-time/space-noise dualities, and derives explicit determinantal formulas for fixed-time and fixed-boundary observables via Fredholm determinants with carefully constructed kernels. Key contributions include Johansson’s and Schütz’s determinant formulas, kernel mappings across model transitions, and rigorous limits that connect Geometric to Exponential to Brownian LPP, yielding corresponding tasep descriptions and Warren-type results. The work consolidates the KPZ-related connections between directed polymers and particle systems, providing deterministic identities and probabilistic tools to study fluctuations and limit shapes in (1+1)-dimensional growth.

Abstract

We explore the connection between tasep-like interacting particle systems and last passage percolation type polymer models, focusing on three models: Geometric, Exponential and Brownian last passage percolation and their associated tasep particle systems. We explain how formulas for certain natural observables in last passage percolation translate to formulas for tasep, by going through a notion of "duality". In turn, we obtain determinantal formulas for last passage percolation with a deterministic boundary and for tasep with a deterministic first particle trajectory.

On the duality between particles and polymers

TL;DR

The paper establishes a comprehensive duality between tasep-like particle systems and last passage percolation (LPP) polymers across Geometric, Exponential, and Brownian settings, using a variational (Hopf–Lax–Oleinik–style) framework. It develops a language and topology for LPP models, builds isomorphisms and space-time/space-noise dualities, and derives explicit determinantal formulas for fixed-time and fixed-boundary observables via Fredholm determinants with carefully constructed kernels. Key contributions include Johansson’s and Schütz’s determinant formulas, kernel mappings across model transitions, and rigorous limits that connect Geometric to Exponential to Brownian LPP, yielding corresponding tasep descriptions and Warren-type results. The work consolidates the KPZ-related connections between directed polymers and particle systems, providing deterministic identities and probabilistic tools to study fluctuations and limit shapes in (1+1)-dimensional growth.

Abstract

We explore the connection between tasep-like interacting particle systems and last passage percolation type polymer models, focusing on three models: Geometric, Exponential and Brownian last passage percolation and their associated tasep particle systems. We explain how formulas for certain natural observables in last passage percolation translate to formulas for tasep, by going through a notion of "duality". In turn, we obtain determinantal formulas for last passage percolation with a deterministic boundary and for tasep with a deterministic first particle trajectory.
Paper Structure (56 sections, 26 theorems, 242 equations)

This paper contains 56 sections, 26 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Duality
  3. Skorokhod reflection
  4. Brownian last passage percolation and Brownian tasep
  5. A formula for the law of Brownian last passage percolation with boundary
  6. The narrow wedge boundary
  7. The flat boundary and random matrices
  8. Brownian tasep
  9. Continuous time tasep
  10. Further context
  11. A soupçon of language
  12. Domains for space, time and noise
  13. Environment and Noise
  14. Directed paths
  15. Last passage percolation
  16. ...and 41 more sections

Key Result

Proposition 1.1

Let $B(t)$ be a standard Brownian motion and $b: [0,\infty) \to \mathbb{R}$ be continuous with $b(0)=0$. Let $0 < t_1 < t_2 < \ldots < t_k$. Then, The determinant is the Fredholm determinant understood through its series expansion. The integral kernels $Q$ and $R$ are as follows. Here $W : [0,\infty) \to \mathbb{R}$ is a standard Brownian motion and $\tau = \inf \{t \geq 0: W(t) \leq b(t)\}$ is

Theorems & Definitions (44)

  • Proposition 1.1
  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • Proposition 3.1
  • proof
  • Theorem 3
  • ...and 34 more