Bounded Discrete Bridges Analysis

• Bounded discrete bridges are constrained lattice paths that remain within fixed bounds over a finite interval and are conditioned to start and end at specified points, with applications in probability, combinatorics, and physics.
• The analysis rigorously corrects previous root dominance assumptions by deploying precise contour deformation and singularity analysis to derive a Rayleigh law for the maximal height.
• Extensions include higher-order asymptotic expansions for Łukasiewicz bridges and periodic walks, offering validated algorithms and explicit error control for practical computational applications.

A bounded discrete bridge is a constrained lattice path, typically a discrete-time random walk or a related process, that is restricted to remain within prescribed (often finite) spatial bounds over a finite time interval, and is conditioned to start and end at specified points. These objects arise naturally in enumerative combinatorics, probability theory, statistical physics, and network theory, and admit a variety of rigorous characterizations and powerful analytical methods. Recent work on the subject has focused on correcting previous incomplete arguments, extending the framework to periodic walks, and delivering precise asymptotic laws for observables such as the maximal height of the bridge.

1. Rigorous Proof of the Limiting Rayleigh Law for Maximal Height

The principal asymptotic result for the maximal height of bounded discrete bridges is that, after suitable centering and scaling, the distribution converges to a Rayleigh law. This limit was asserted by Banderier & Nicodeme (2010) based on asymptotic kernel methods, but their proof used root dominance properties that do not hold globally in the complex plane. The corrected proof restricts attention to a real interval $(0, \rho)$ for the kernel variable $z$, within which definite dominance relations for the roots hold and integration contours can be chosen so that the asymptotics are valid.

Let $P(u)$ denote the characteristic (Laurent) polynomial of the walk, and consider the generating function arising from the kernel equation $1 - zP(u) = 0$. The critical roots $u_1(z)$ (dominant small root) and $v_1(z)$ (dominant large root) play a central role. The corrected analysis, building upon the Recognized Domination Lemma ([Banderier–Flajolet 2002, Lemma 2]), shows that

$$\lim_{n \to \infty} \Pr\left(\mathrm{height\ of\ bridge\ }> x \sigma \sqrt{n}\right) = \exp\left( -2 x^2 \frac{\rho}{\tau^2} \right)$$

where $\tau$ is determined by $P'(\tau) = 0$, $\rho = 1/P(\tau)$, and $\sigma^2 = P''(\tau)$. The limiting law is Rayleigh, and this asymptotic holds for general aperiodic walks whose kernel polynomial has simple roots in the complex field. The proof tracks the roots' behavior only along those real segments where the dominance property remains valid, deploying precise contour deformation and singularity analysis in the sense of Flajolet and Sedgewick.

2. Asymptotic Expansions for Łukasiewicz Bridges and Higher-Order Corrections

For Łukasiewicz bridges—lattice paths where the allowed negative jump is $-1$ and the kernel is suitably simple—the paper supplies higher-order asymptotic expansions, not just leading-order results. The expansion for the probability that the maximal height exceeds $h = x \sigma \sqrt{n}$ is

$$\beta_n^{>h} = e^{-2x^2} \left( 1 + \frac{f_1(x)}{\sqrt{n}} + \frac{f_2(x)}{n} + O(n^{-3/2}) \right)$$

where the coefficients $f_1(x)$ and $f_2(x)$ are explicitly provided in terms of Hermite polynomials evaluated at $4x$, and depend upon higher moments $P''(1), P'''(1), P^{(4)}(1)$ of the walk's step distribution. Detailed numerical comparisons for Dyck paths (undirected (+1,-1) bridges) exhibit excellent agreement with these formulas, confirming their predictive power for finite but large $n$. The formulas are algorithmically tractable, supporting symbolic computation of higher-order terms by Newton iteration and recursive use of generating function technology.

3. Extension to Periodic Walks and Decomposition of the Kernel

This work broadens the theory to cover periodic bridges, those for which the support of the walk's increment distribution is not coprime and the characteristic polynomial decomposes over $\mathbb{C}$ as $u^c P(u) = H(u^p)$, with $p$ (the period) the gcd of the exponents in the support. In this setting, singularities associated to the kernel roots are replicated according to the period, and the Hankel-type contours required for singularity extraction are accordingly multiplied. For bridges of lengths divisible by the period $p$, the partitioned contributions from the dominant singularities sum in the limit, and the probability formulas retain a Rayleigh-type limiting law, with the period factor systematically canceling in normalized ratios.

A crucial assumption throughout is that $P(u)$ decomposes with no repeated roots, ensuring that all singularities are simple and the local analysis near each dominant singularity remains tractable. Integration contours are carefully constructed (e.g., see Lemma~\ref{lem:pdomin}) to remain in domains where root domination is valid.

4. Root Dominance and Precise Domain of Validity

A salient technical point is that kernel root dominance—the property that the moduli of $u_1(z)$ and $v_1(z)$ control all other roots—only holds on the positive real segment $(0, \rho)$ (for aperiodic walks) or within suitable conic sectors (for periodic walks). It fails for the full disk $|z| < \rho$, with explicit counterexamples identified (cf. Wallner, 2018). The corrected proof constructs the required Cauchy and Hankel contours narrowly within these domains, strictly avoiding those regions where root crossings might subvert the dominance ordering. This restriction is essential for rigorous error bounds, ensuring that exponentially small contributions from non-dominant roots do not pollute the main terms in the asymptotics.

5. Comparative Advances: Improved Rigor, Extensions, and Validation

A key advancement of this paper over Banderier & Nicodeme (2010) is the rigorous restriction and careful handling of root domination, which was previously assumed to hold globally. The work not only corrects the primary proof but supplies detailed treatment of periodic walks, delivers explicit and arbitrary-order asymptotic expansions for special bridge classes (notably Łukasiewicz), and validates these expansions against both simulations and known exact solutions (such as those derived from the reflection principle for Dyck paths). Error terms are carefully quantified, and the analysis shows absolute error is at worst $O(1/\sqrt{n})$ with higher-order corrections explicit.

Additionally, the discussion connects these precise discrete results to probabilistic strong embedding theorems (Komlós–Major–Tusnády, Chatterjee), asserting tight coupling between bounded discrete bridges and Brownian motion. This connection substantiates the observed universality and smooth transition to continuous path asymptotics in the scaling limit.

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Summary Table: Key Improvements and Distinctions

Aspect	Banderier & Nicodeme (2010)	Nicodème (2024)
Use of root dominance	Assumed on $|z| < \rho$ (incorrect)	Correctly restricted to real intervals
Periodic walks	Not analyzed	Full periodic treatment provided
Higher-order expansions	Limited	Explicit, symbolic expansions provided
Numerical & analytic validation	Partial	Exhaustive and high-precision
Error control	Unspecified or broad	Explicit error quantification
Brownian/Strong embedding relation	Claimed	Directly analyzed, tight coupling shown

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By precisely localizing analytical tools to their valid domains and systematically expanding the asymptotic theory, bounded discrete bridges—especially those of periodic type—are placed on firm mathematical footing. The work yields high-precision formulae for the probabilities of rare events (such as excursion heights), explicit error bounds, and robust algorithms for high-fidelity enumeration and simulation of lattice-constrained stochastic paths.