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Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases

Erik Brodsky, Eva R. Engel, Connor Panish, Lillian Stolberg

TL;DR

The paper investigates two complementary routes to generate dynamics for the Type D ASEP with two particle classes: a probabilistic construction via stochastic fusion and an algebraic construction from a Casimir element of $U_q(\mathfrak{so}_6)$ using crystal bases. It demonstrates a fundamental mismatch between the two resulting generators, namely a 9-state-per-site probabilistic process and a 14-state-per-site algebraic process, and shows that the Type A correspondence does not extend to all finite-dimensional simple Lie algebras. The probabilistic analysis provides detailed block decompositions, limits as drift grows, and reversible measures, while the algebraic analysis builds a 196×196 fused generator from $W\otimes W$ and a ground-state transformation, revealing structural differences with the probabilistic generator. Collectively, these results clarify the limitations of generalizing stochastic-fusion/gnd-state correspondences beyond Type A and offer a precise algebraic framework for comparing the two approaches in Type D.

Abstract

The Type D asymmetric simple exclusion process (ASEP) is a particle system involving two classes of particles that can be viewed from both a probabilistic and an algebraic perspective (arXiv:2011.13473). From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP and analyze the outcome on generator matrices, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of $U_q(so_6)$, using crystal bases to analyze and manipulate various representations of $U_q(so_6)$. We conclude that both approaches produce different processes and therefore the previous method of arXiv:1908.02359, which analyzed the usual ASEP, does not generalize to all finite-dimensional simple Lie algebras.

Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases

TL;DR

The paper investigates two complementary routes to generate dynamics for the Type D ASEP with two particle classes: a probabilistic construction via stochastic fusion and an algebraic construction from a Casimir element of using crystal bases. It demonstrates a fundamental mismatch between the two resulting generators, namely a 9-state-per-site probabilistic process and a 14-state-per-site algebraic process, and shows that the Type A correspondence does not extend to all finite-dimensional simple Lie algebras. The probabilistic analysis provides detailed block decompositions, limits as drift grows, and reversible measures, while the algebraic analysis builds a 196×196 fused generator from and a ground-state transformation, revealing structural differences with the probabilistic generator. Collectively, these results clarify the limitations of generalizing stochastic-fusion/gnd-state correspondences beyond Type A and offer a precise algebraic framework for comparing the two approaches in Type D.

Abstract

The Type D asymmetric simple exclusion process (ASEP) is a particle system involving two classes of particles that can be viewed from both a probabilistic and an algebraic perspective (arXiv:2011.13473). From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP and analyze the outcome on generator matrices, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of , using crystal bases to analyze and manipulate various representations of . We conclude that both approaches produce different processes and therefore the previous method of arXiv:1908.02359, which analyzed the usual ASEP, does not generalize to all finite-dimensional simple Lie algebras.
Paper Structure (48 sections, 40 theorems, 254 equations, 7 figures, 2 tables)

This paper contains 48 sections, 40 theorems, 254 equations, 7 figures, 2 tables.

Table of Contents

  1. Abstract
  2. Introduction
  3. Probability Notation
  4. Type D ASEP
  5. Stochastic Fusion
  6. Transition and Generator Matrices
  7. Algebraic Notation
  8. Probabilistic Results
  9. Producing Stochastic Fusion Matrix
  10. Taking n to Infinity and Spectral Gaps
  11. Reversible Measures
  12. Markov Duality
  13. Duality with Quantum q-Krawtchouk Polynomials
  14. Other Avenues to Duality
  15. Algebraic Results
  16. ...and 33 more sections

Key Result

Theorem 3.1.0

The generator for Type D ASEP on $K/2$$\gamma$-lattice sites, denoted $L^{(K)}_Q$, is equal to $\Lambda^{(K)} L^{(K)}_m \Phi^{(K)}$.

Figures (7)

  • Figure 1: The fusion states for one $\gamma$ site from left to right $\langle 0 \rangle, \langle 1 \rangle, \langle 2 \rangle,\langle 3 \rangle, \langle 11 \rangle,\langle 22 \rangle,\langle 31 \rangle,\langle 32 \rangle,\langle 33 \rangle$
  • Figure 2: Depiction of fusion mapping $(3,1,0,2) \rightarrow \langle 31,2 \rangle$ and fission mapping $\langle 31,2 \rangle \rightarrow (3,1,0,2)$.
  • Figure 3: Crystal graph of fundamental representation $\mathcal{U}_q(\mathfrak{so}_6)$
  • Figure 4: Crystal graph of $V(2L_1)$ in the representation $V \otimes V$
  • Figure 5: Crystal graph of $\mathcal{U}_q(\mathfrak{so}_6)$ in the representation $V \otimes V$
  • ...and 2 more figures

Theorems & Definitions (82)

  • Definition 2.1.1
  • Definition 2.2.1
  • Definition 2.2.2
  • Definition 2.2.3
  • Definition 2.2.4
  • Definition 2.2.5
  • Definition 2.2.6
  • Theorem 3.1.0
  • Corollary 3.1.1
  • Proposition 3.1.1
  • ...and 72 more