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Limits of Random Motzkin paths with KPZ related asymptotics

Wlodzimierz Bryc, Alexey Kuznetsov, Jacek Wesolowski

TL;DR

We analyze boundary fluctuations of $q$-weighted Motzkin paths with random endpoints. A matrix- and orthogonal-polynomial framework driven by Al-Salam–Chihara polynomials yields a general boundary-limit theorem, producing two independent Markov chains for the initial and final segments. In macroscopic limits, one regime ($q$ fixed, $\rho_0\to1$ at a $\sqrt{N}$-rate) converges to a 3D Bessel-type process, while a joint $q\to1$ scaling (with appropriate centering) leads to a KPZ-half-line–type Markov process described by the Yakubovich kernel and the $K_0$ spectral weight; these connections illuminate non-Gaussian boundary fluctuations in KPZ on the half-line. The analysis hinges on sharp endpoint asymptotics for the Al-Salam–Chihara polynomials and precise $q$-special-function limits (via $q$-Pochhammer, $q$-Gamma and theta-function techniques). Overall, the work bridges discrete Motzkin-path models with KPZ-half-line stationary measures and conjectured fixed-point structures through rigorous boundary asymptotics.

Abstract

We study Motzkin paths of length $L$ with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as $L$ becomes large. We then study macroscopic limits of the resulting processes, where in two different regimes we obtain Markov processes that appeared in the description of the stationary measure for the KPZ equation on the half line and of conjectural stationary measure of the hypothetical KPZ fixed point on the half line. Our results rely on the behavior of the Al-Salam--Chihara polynomials in the neighbourhood of the upper end of their orthogonality interval and on the limiting properties of the $q$-Pochhammer and $q$-Gamma functions as $q\nearrow 1$.

Limits of Random Motzkin paths with KPZ related asymptotics

TL;DR

We analyze boundary fluctuations of -weighted Motzkin paths with random endpoints. A matrix- and orthogonal-polynomial framework driven by Al-Salam–Chihara polynomials yields a general boundary-limit theorem, producing two independent Markov chains for the initial and final segments. In macroscopic limits, one regime ( fixed, at a -rate) converges to a 3D Bessel-type process, while a joint scaling (with appropriate centering) leads to a KPZ-half-line–type Markov process described by the Yakubovich kernel and the spectral weight; these connections illuminate non-Gaussian boundary fluctuations in KPZ on the half-line. The analysis hinges on sharp endpoint asymptotics for the Al-Salam–Chihara polynomials and precise -special-function limits (via -Pochhammer, -Gamma and theta-function techniques). Overall, the work bridges discrete Motzkin-path models with KPZ-half-line stationary measures and conjectured fixed-point structures through rigorous boundary asymptotics.

Abstract

We study Motzkin paths of length with general weights on the edges and end points. We investigate the limit behavior of the initial and final segments of the random Motzkin path viewed as a pair of processes starting from each of the two end points as becomes large. We then study macroscopic limits of the resulting processes, where in two different regimes we obtain Markov processes that appeared in the description of the stationary measure for the KPZ equation on the half line and of conjectural stationary measure of the hypothetical KPZ fixed point on the half line. Our results rely on the behavior of the Al-Salam--Chihara polynomials in the neighbourhood of the upper end of their orthogonality interval and on the limiting properties of the -Pochhammer and -Gamma functions as .
Paper Structure (14 sections, 24 theorems, 221 equations, 2 figures)

This paper contains 14 sections, 24 theorems, 221 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. A general limit theorem for random Motzkin paths at the boundary
  3. Matrix representation and integral representation
  4. Limit theorem
  5. Properties of the Al-Salam--Chihara polynomials
  6. Maximum of the Al-Salam--Chihara polynomials
  7. Pointwise asymptotics near the end of the interval of orthogonality
  8. Motzkin paths with $q$-number weights
  9. Local limit theorems
  10. Proofs of Theorems \ref{['Thm:2']} and \ref{['Thm:3']}
  11. Auxiliary results on special functions
  12. Some useful limits for $q\nearrow 1$
  13. Bounds on $\Gamma_q$
  14. Additional technical lemmas

Key Result

Theorem 1.1

Suppose that $0\leq \rho_0,\rho_1,q<1$ and $0<\sigma \leq 1$. Then as $L\to\infty$, where on the left-hand side we have random Motzkin paths with respect to measure Pr0 and on the right-hand side we have two independent Markov chains ${\boldsymbol X}=\left\{X_k\right\}_{k\ge 0}$, ${\boldsymbol Y}=\left\{Y_k\right\}_{k\ge 0}$ with the same transition probabilities o for $n=0,1,\ldots$ and with the

Figures (2)

  • Figure 1: Motzkin path ${\boldsymbol \gamma}=(2,1,1,0,1,1,2,1,0,1)\in\mathcal{M}^{(9)}$ with weight contributions marked at the edges. The probability of selecting the path shown above from $\mathcal{M}^{(9)}$ is $\mathds{P}_9({\boldsymbol \gamma})=\frac{\alpha_2 \beta_1}{\mathfrak C_{{\boldsymbol \alpha},{\boldsymbol \beta}, 9} } b_1^2 a_0^2 a_1 c_1^2 c_2^2$.
  • Figure 2: Domain $D_{\epsilon}$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: matrix ansatz
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 40 more