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Balanced weighted Motzkin paths: Pearson structure and saddlepoint asymptotics

Alexander Omelchenko

TL;DR

The paper analyzes balanced Motzkin paths with linearly varying step weights, deriving a Pearson-type PDE for the exponential generating function and solving it via characteristics across all drift regimes. A moving algebraic singularity τ(x) governs both local Gaussian behaviour and the global large-deviation profile, with a coalescing-saddle structure that yields precise finite-n asymptotics. A uniform finite-n approximation is established through Daniels lattice saddlepoint methods, producing a single formula that remains accurate from the center through the tails. In the large-n limit, the limit cumulant generating function F(θ) = log τ(1) − log τ(e^θ) drives a strictly convex rate function I(u) via Legendre transform, delivering an n-speed large-deviation principle for K_n/n and a coherent global picture that extends to related weighted path models and tridiagonal recurrences.

Abstract

We analyse weighted Motzkin paths with step multiplicities that vary linearly with height. In the balanced case the associated exponential generating function satisfies a Pearson-type PDE, and solving by characteristics yields closed expressions in all drift regimes. These formulas reveal a moving algebraic singularity that governs both local and global behaviour. Locally this gives a Gaussian central window for the terminal-height distribution, while globally we identify an explicit limit cumulant generating function and prove an $n$-speed large-deviation principle. For finite $n$, Daniels' lattice saddlepoint approximation provides a single formula that is accurate across the full range of $k$; in all quadratic regimes it achieves a uniform interior relative error of order $n^{-1}$. The results link Pearson geometry with uniform saddlepoint methods and extend naturally to other weighted path models and tridiagonal recurrences.

Balanced weighted Motzkin paths: Pearson structure and saddlepoint asymptotics

TL;DR

The paper analyzes balanced Motzkin paths with linearly varying step weights, deriving a Pearson-type PDE for the exponential generating function and solving it via characteristics across all drift regimes. A moving algebraic singularity τ(x) governs both local Gaussian behaviour and the global large-deviation profile, with a coalescing-saddle structure that yields precise finite-n asymptotics. A uniform finite-n approximation is established through Daniels lattice saddlepoint methods, producing a single formula that remains accurate from the center through the tails. In the large-n limit, the limit cumulant generating function F(θ) = log τ(1) − log τ(e^θ) drives a strictly convex rate function I(u) via Legendre transform, delivering an n-speed large-deviation principle for K_n/n and a coherent global picture that extends to related weighted path models and tridiagonal recurrences.

Abstract

We analyse weighted Motzkin paths with step multiplicities that vary linearly with height. In the balanced case the associated exponential generating function satisfies a Pearson-type PDE, and solving by characteristics yields closed expressions in all drift regimes. These formulas reveal a moving algebraic singularity that governs both local and global behaviour. Locally this gives a Gaussian central window for the terminal-height distribution, while globally we identify an explicit limit cumulant generating function and prove an -speed large-deviation principle. For finite , Daniels' lattice saddlepoint approximation provides a single formula that is accurate across the full range of ; in all quadratic regimes it achieves a uniform interior relative error of order . The results link Pearson geometry with uniform saddlepoint methods and extend naturally to other weighted path models and tridiagonal recurrences.
Paper Structure (7 sections, 5 theorems, 78 equations, 5 figures)

This paper contains 7 sections, 5 theorems, 78 equations, 5 figures.

Table of Contents

  1. Motzkin paths with linear weights: model, recurrence, and closed forms
  2. From weights to probabilities and local asymptotics
  3. A uniform finite-$n$ approximation via the discrete saddlepoint
  4. From finite--$n$ cumulants to a limit rate profile
  5. Numerical computation and results
  6. Local asymptotics in the linear regime ($A=0$)
  7. Local asymptotics in the quadratic regime ($A>0$)

Key Result

Proposition 2.1

Fix $x$ in the real domain of analyticity containing $x=1$, and suppose that near its smallest positive singular time $\tau(x)$ the balanced generating function admits the local representation with $\mathcal{A}(x)\mathrel{\not=} 0$ analytic. Then, as $n\to\infty$, Writing $\chi(x):=-\tau'(x)/\tau(x)$ and $S_n=P_n(1)$ one has and, in the central window $k=\mu_n+O(\sigma_n)$, the Gaussian local l

Figures (5)

  • Figure 1: The classical Motzkin triangle (a) and a triangle with linearly varying multiplicities (b).
  • Figure 2: The classical Dyck triangle (a) and a triangle with linearly varying multiplicities (b).
  • Figure 3: Two real roots ($\Delta>0$), linear scale. Exact distribution (dots), Gaussian local approximation (dashed), and Daniels saddlepoint curve (solid). Parameters: $A=1$, $B=6$, $C=5$, $\alpha_0=8$, $\gamma_0=1$, $n=100$.
  • Figure 4: Two real roots ($\Delta>0$), logarithmic scale. Same three curves as in Figure \ref{['fig:profile-DeltaPos-lin']}, with large-deviation line $-nI(u)/\log 10$ (dotted).
  • Figure 5: Large-deviation scaling for $\Delta>0$. Points show $-\tfrac{1}{N}\log p_{N,\lfloor uN\rfloor}$. The solid curve is the rate function $I(u)$.

Theorems & Definitions (11)

  • Proposition 2.1: Coalescing algebraic singularity
  • proof : Proof outline
  • Theorem 3.1: Uniform finite--$n$ accuracy
  • proof : Proof idea
  • Lemma 4.1: Existence and formula for the limit CGF
  • proof
  • Lemma 4.2: Smoothness, strict convexity, and range of $F'$
  • proof
  • Remark 4.3: On the domain in $\vartheta$
  • Theorem 4.4: Large deviations and the rate function
  • ...and 1 more