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Free fermionic probability theory and K-theoretic Schubert calculus

Shinsuke Iwao, Kohei Motegi, Travis Scrimshaw

TL;DR

This work bridges four discrete-time TASEP variants with K-theoretic Schubert calculus by encoding particle dynamics in free-fermion language and refining Schur operators to act on partitions. It proves that the n-step transition kernels align with dual refined Grothendieck polynomials (and their weak variants) up to simple normalization, providing both operator-based and bijective proofs. The authors derive determinant formulas for multi-point distributions, continuous-time limits, and introduce a canonical Grothendieck-parameterized process that highlights a local-current interpretation. The results reveal a deep link between interacting particle systems and K-theoretic symmetric functions, enabling combinatorial proofs and potentially new geometric insights with implications for percolation models and Schubert calculus.

Abstract

For each of the four particle processes given by Dieker and Warren [arXiv:0707.1843], we show the $n$-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.

Free fermionic probability theory and K-theoretic Schubert calculus

TL;DR

This work bridges four discrete-time TASEP variants with K-theoretic Schubert calculus by encoding particle dynamics in free-fermion language and refining Schur operators to act on partitions. It proves that the n-step transition kernels align with dual refined Grothendieck polynomials (and their weak variants) up to simple normalization, providing both operator-based and bijective proofs. The authors derive determinant formulas for multi-point distributions, continuous-time limits, and introduce a canonical Grothendieck-parameterized process that highlights a local-current interpretation. The results reveal a deep link between interacting particle systems and K-theoretic symmetric functions, enabling combinatorial proofs and potentially new geometric insights with implications for percolation models and Schubert calculus.

Abstract

For each of the four particle processes given by Dieker and Warren [arXiv:0707.1843], we show the -step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.
Paper Structure (26 sections, 32 theorems, 232 equations, 5 figures, 2 tables)

This paper contains 26 sections, 32 theorems, 232 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. K-theoretic symmetric functions
  4. Supersymmetric functions
  5. Free fermions and Schur operators
  6. Particle processes
  7. Schur operators for canonical Grothendieck polynomials
  8. Noncommutative blocking operators
  9. Noncommutative pushing operators
  10. Operator dynamics
  11. Pushing operators
  12. Blocking operators
  13. Bijective description
  14. Case A: Geometric pushing
  15. Case D: Bernoulli pushing
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose $\ell(\lambda), \lambda_1 \leq \ell$. Suppose $\pi_j x_i \in (0, 1)$ and $\rho_j x_i > 0$ for all $i$ and $j$. Set $\alpha_j = \rho_{j+1}$ and $\beta_j = \pi_{j+1}$. The transition probabilities of the four particle systems $X=X^\mathrm{A},X^{\mathrm{B}}, X^\mathrm{C}, X^\mathrm{D}$ from the

Figures (5)

  • Figure 1: Examples of the third particle making a jump of $6$ steps with the pushing (left) and blocking (right) behaviors.
  • Figure 2: Samples of the continuous time limit TASEP with $\ell = 500$ particles at time $t = 500$ with rate $\boldsymbol{\pi} = 1$ under the blocking (left) and pushing (right) behavior.
  • Figure 3: Samples of TASEP with $\ell = 500$ particles with $\boldsymbol{\pi} = 1$ and $\mathbf{x} = p = 0.01$ with $n = \lfloor 500 / p \rfloor$ under the blocking behavior (left) and pushing behavior (right).
  • Figure 4: A sampling using $10000$ samples of the modified geometric distribution $\mathsf{P}_{\mathcal{G}}$ for $x_i = 1$, $\pi_j = .5$, and $\alpha_k = 1 - k e^{-k/2}$ (blue) under the exact distribution (red), which is under the geometric distribution (green).
  • Figure 5: Samples of blocking TASEP with $\ell = 500$ particles after $n = 50000$ time steps with (left) $\boldsymbol{\pi} = 1$, $\mathbf{x} = 0.01$, and $\boldsymbol{\alpha} = -0.5$; (right) $\boldsymbol{\pi} = 0.5$, $\mathbf{x} = .2$, and $\alpha_k = 0.5 \sin(k/50)^6$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 2.1: HJKSS24
  • Proposition 2.2
  • Proposition 2.3: Branching rules IMS22
  • Theorem 2.4: Skew Cauchy formula
  • Theorem 2.5: FG98
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • Example 3.1
  • ...and 65 more