Gaussian Random Walks (GRW)

• Gaussian Random Walk (GRW) is a discrete-time process where each step is defined by a Gaussian distribution, leading to classical diffusion and potential superdiffusion.
• Variants such as Correlated GRWs and discrete models (e.g., Jacobi walks) adjust step correlations and distributions to capture persistence, clustering, and rapid convergence in algorithms.
• Methodologies using GRW, including permutation-based (RPST) constructions, extend to algorithmic applications in discrepancy minimization and uncertainty modeling with practical computational guarantees.

A Gaussian random walk (GRW) is a discrete-time stochastic process in which each step's increment is governed by a Gaussian (normal) distribution. GRWs serve as canonical models for diffusion, uncertainty propagation, and algorithmic randomization in mathematics, physics, and computer science. Their multivariate and correlated variants, along with generalizations motivated by combinatorial and information-theoretic considerations, reveal a spectrum of behavior ranging from classical diffusion to persistent superdiffusion and sophisticated evidence-reasoning frameworks.

1. Formal Definition and Variants

Let $S_n = \sum_{k=1}^n x_k$ denote the state of a random walk after $n$ steps in $\mathbb{R}^d$. The prototypical GRW is the Standard Gaussian Random Walk (SGRW), where $\{x_k\}$ are i.i.d. $\mathcal{N}(0, \sigma^2 I_d)$, yielding

$$p_{\mathrm{SGRW}}(x_k) = \frac{1}{(2\pi\sigma^2)^{d/2}} \exp\left(-\frac{|x_k|^2}{2\sigma^2}\right)$$

and for the displacement after $n$ steps,

$$p_n^{(\mathrm{SGRW})}(x) = \frac{1}{\sqrt{2\pi n \sigma^2}} \exp\left(-\frac{x^2}{2 n \sigma^2}\right)$$

in one dimension. The process is Markovian, memoryless, and exhibits Gaussian scaling of displacements, $\langle S_n^2\rangle = n\sigma^2$ (Bagarti, 2011).

Correlated Gaussian Random Walks (CGRW) generalize this by allowing increments $x_k$ to depend on prior steps. In the CGRW framework,

$x_k \sim \mathcal{N}(\mu(x_{k-1}), \sigma^2 I_d)$

with $\mu(x_{k-1})$ encoding directional or magnitude correlations. The classical SGRW is recovered for $\mu \equiv 0$. Analytical and numerical treatments reveal regimes of persistence and clustering as correlation structure is varied (Bagarti, 2011).

Alternative GRW constructions, such as the Gaussian fixed-point walks, replace Gaussian increment laws with discrete step sets—e.g., $\{0, \pm1\}$ or $\{\pm1, 2\}$—and design transition probabilities to ensure invariance of the Gaussian measure (Liu et al., 2021).

2. Step Distributions and Correlation Structures

In the SGRW, increments are uncorrelated Gaussians. For the CGRW, increments can exhibit directional persistence (steps tend to continue in the same direction) or more complex dependencies. For a model with $\mu(x_{k-1}) = \mathrm{sgn}(x_{k-1})$ in one dimension, the process is more likely to persist in its previous direction, resulting in a bimodal endpoint distribution in the small-noise limit. Another model includes magnitude correlation, where

$$\mu(x) = \begin{cases} \mathrm{sgn}(x)p, & |x|\le x_0 \ 0, & |x|>x_0 \end{cases}$$

and produces oscillatory PDFs reflecting “stop‐and‐go” dynamics (Bagarti, 2011).

Discrete GRW variants designed to preserve Gaussian stationary measures use transition kernels derived from theta-function identities. For instance, the “Jacobi walk” uses steps in $\{0,\pm1\}$ with transition probabilities $p_\sigma(x)$ and $r_\sigma(f)$, ensuring detailed balance with the discrete Gaussian measure

$$\pi_f(n+f) = Z_f^{-1} \exp\left(-\frac{(n+f)^2}{2\sigma^2}\right)$$

across shifted lattices $S_f = \{n+f: n \in \mathbb{Z}\}$ (Liu et al., 2021).

3. Limiting Behavior and Functional Central Limit Theorems

For any GRW whose increments are i.i.d. with finite second moments and mean zero, the multivariate central limit theorem (CLT) ensures that the scaled displacement converges to a Gaussian law: $$\frac{S_m}{\sqrt{m}} \xrightarrow{d} \mathcal{N}(0, \sigma^2 I)$$ For processes such as those constructed within the Random Permutation Set Theory (RPST) framework, the CLT obtains even though each finite-$n$ increment is non-Gaussian; only asymptotically and in aggregate do the sums become Gaussian (Zhou et al., 5 Apr 2024). Taking the diffusive limit leads to a functional CLT (Donsker’s theorem), yielding

$$W_m(t) = \frac{S_{\lfloor mt \rfloor}}{\sqrt{m \sigma^2}} \xrightarrow{d} W(t)$$

where $W(t)$ is a standard $d$-dimensional Wiener process.

4. Alternative Constructions and Connections to Uncertainty Frameworks

The RPST approach constructs random walks by sampling permutations of $n$ elements, mapping their components onto radial vectors in $\mathbb{R}^2$ and summing to obtain increment vectors $X_k$. With a maximum‐entropy permutation mass function, as $n \to \infty$ and appropriate scaling, the random walk converges in law to a Wiener process (Zhou et al., 5 Apr 2024).

These constructive models highlight combinatorial mechanisms (symmetrization, entropy maximization) as alternatives to Gaussian sampling. The physical interpretation links the entropy limit $e \cdot (n!)^2$ with the mean-square displacement of Brownian motion. The RPST framework extends GRW methodologies into domains characterized by structural uncertainty and belief function calculus, embedding classical stochastic calculus in a more extensive evidence-theoretic setting (Zhou et al., 5 Apr 2024).

Discrete GRW models, such as those by Jacobi and Ramanujan walks, provide analytically explicit, finite-step Markov chains whose unique stationary distribution is Gaussian. A crucial property is that at least 96% of steps are nonzero ($\pm1$), essential for rapid convergence in partial-coloring algorithms, with transition probabilities bounded using the Jacobi triple-product (Liu et al., 2021).

5. Algorithmic and Application Domains

GRWs underpin algorithmic applications in online discrepancy minimization, vector balancing, and randomized rounding. Algorithms resampling partial colorings correspond directly to GRW increments, with theoretical guarantees on sup-norm discrepancy and high frequency of “active” steps. In the discrete fixed-point GRW models,

• Online partial coloring algorithms output $\epsilon_i \in \{-1, 0, +1\}$ (“skip” or sign), ensuring $$\left\|\sum_{i=1}^{\ell} \epsilon_i v_i\right\|_\infty$$ is $O(\sqrt{\log(nt/\delta)\log(t/\delta)})$ with failure probability $\leq \delta$ and at least 96% nonzero $\epsilon_i$ (Liu et al., 2021).
• Komlós-type bounds are obtained in linear or nearly input-sparsity time, and restart/recolor arguments allow conversion to fully $\pm1$ signings (Liu et al., 2021).

In ecological modeling, CGRWs enable analytical and numerical exploration of animal dispersal, where persistence and clustering phenomena are captured by adjusting $\mu(x_{k-1})$; simulations confirm analytic and scaling predictions (Bagarti, 2011).

6. Comparison Table: GRW Variants

Model	Increment Law	Long-Time Law
SGRW	i.i.d. $\mathcal{N}(0, \sigma^2 I_d)$	Gaussian, diffusive
CGRW	$\mathcal{N}(\mu(x_{k-1}), \sigma^2 I_d)$	Diffusive, superdiffusive, or clustered, depending on $\mu$
RPST-RGW [Editor’s term]	Permutation-based, non-Gaussian	Gaussian via CLT; Wiener in scaling limit
Jacobi/Ramanujan walks	Discrete with $\{0,\pm1\}$, $\{\pm1,2\}$	Stationary Gaussian, high $\pm1$ frequency

In all cases, key parameters are increment correlation structure and distributional properties. Only the SGRW features strictly i.i.d. Gaussian increments. others, via construction or correlation, achieve Gaussian diffusion properties either in the aggregate or at stationarity.

7. Research Directions and Theoretical Insights

Current GRW research elucidates the relationship between combinatorial structures, entropy, and diffusion, as demonstrated by RPST-based models (Zhou et al., 5 Apr 2024). The equivalence between certain discrete Markov chains and continuous Gaussian laws points to deep connections between special functions, invariance principles, and random walks (Liu et al., 2021). Analytical tractability of these processes has facilitated advances in algorithmic discrepancy theory and evidence reasoning.

A plausible implication is that further exploration of permutation-based and correlated increment structures could yield new classes of stochastic processes with tailored diffusion and uncertainty properties, relevant for high-dimensional inference, molecular simulation, and information theory. Correlation does not invariably imply directional persistence; as demonstrated in (Bagarti, 2011), the specific form of step dependence determines whether a process is diffusive, superdiffusive, or exhibits clustering behavior.