Process-Level Scaling Limits

• Process-level scaling limits are theoretical frameworks that rigorously describe the asymptotic behavior of entire sample paths of stochastic processes under extreme parameter regimes.
• They enable dimensionality reduction by mapping complex, high-dimensional dynamics onto tractable, universal limiting processes observed in queueing systems, particle models, and random matrices.
• These limits provide practical insights into universality phenomena, phase transitions, and computational shortcuts across diverse fields like probability theory, statistical physics, and network algorithms.

Process-level scaling limits characterize the asymptotic behavior of entire sample paths of stochastic processes—such as queues, interacting particle systems, or combinatorial structures—under large-scale or extreme parameter regimes. Rather than focusing on finite-dimensional distributions or static quantities, process-level (or functional) scaling limits rigorously describe the convergence (often in Skorokhod space) of rescaled path-valued observables to limiting (often universal) processes. These results are foundational in establishing universality, dimensionality reduction, and tractable approximations for complex systems across mathematics, probability theory, and statistical physics.

1. Core Concepts and Definitions

At the heart of process-level scaling limits is the paper of families of stochastic processes indexed by a scaling parameter (e.g., system size $n$, intensity $\lambda$, number of particles $d$, or a heavy-traffic parameter $r$) whose time and/or space axes are rescaled so that sample paths converge, in probability or distribution, as the parameter diverges. Typical function spaces used for the convergence are Skorokhod spaces $D([0,T], S)$, often equipped with the $J_1$ topology, or relevant Gromov–Hausdorff–Prokhorov or metric measure spaces for structures like random trees.

The formal task is, given a process $X^{(n)} = (X^{(n)}_t)_{t\geq 0}$ and a suitable rescaling of time and/or space ($X^n_t = a_n^{-1} X^{(n)}_{b_n t}$), to prove weak (distributional) convergence:

$X^n \Longrightarrow X^*$

in $D([0,T], S)$ for some limiting process $X^*$, which characterizes the large-scale dynamics of the original system.

Key motivations include:

• Universality: Many seemingly different systems (e.g., various queueing networks, random planar maps, random matrix eigenvalues) exhibit the same scaling limit.
• Dimensionality reduction: Infinite-dimensional or highly complex systems may, in heavy-traffic or critical regimes, collapse onto tractable low-dimensional or well-understood processes.
• Functional weak convergence: Scaling limit theorems often guarantee convergence of observables and statistics that are (possibly path-dependent) functionals of the original dynamics.

2. Methodological Frameworks and Notable Examples

Queueing Systems and State-Space Collapse

In processor sharing queues with constraints (e.g., the $K$-limited processor sharing or LPS model), the high-dimensional state (e.g., a pair of random counting measures for buffer jobs and jobs in service) is shown—under critical load and diffusion scaling—to "collapse" onto a one-dimensional manifold determined solely by the workload process. Explicitly, after scaling time by $r^2$ and space by $1/r$ (diffusion scaling), the process-level dynamics of these measures are asymptotically captured by a deterministic (lifting) map of a reflected Brownian motion:

$$\Delta_{K,\nu}(w) = \Bigl( \frac{(w - K\beta_e)^+}{\beta} \nu,\; \frac{(w\wedge K\beta_e)}{\beta_e} \nu_e \Bigr)$$

with the system size process converging to a piecewise linear function of the RBM:

$$X^*(t) = \frac{(W^*(t) - K\beta_e)^+}{\beta} + \frac{(W^*(t) \wedge K\beta_e)}{\beta_e}$$

This mechanism is known as state-space collapse (0912.5306).

Coalescent Processes and Ornstein–Uhlenbeck Limits

For block-counting processes in $\Lambda$- or $\Xi$-coalescents, logarithmic rescaling (reflecting the "speed" at which the number of blocks decreases) together with centering leads to convergence toward generalized Ornstein–Uhlenbeck processes. For instance, for a block-counting process $N^{(n)}_t$ starting from size $n$,

$$X^{(n)}_t = \log N^{(n)}_t - e^{-b t} \log n \Longrightarrow X_t$$

where $X$ solves a stochastic differential equation with a negative drift and jump components determined by the driving measure, with detailed convergence results for the generator and Feller semigroup (Möhle et al., 2021, Möhle et al., 2022). Siegmund duality extends these results to fixation lines.

Particle Systems and Superdiffusive Hydrodynamics

In exclusion processes with slow bonds or spatial bottlenecks, the interplay between the number of slow sites ($k$), the site scaling parameter ($n$), bond slowdown exponent ($\beta$), and time acceleration can produce various macroscopic regimes. When time is speeded up superdiffusively ($k^2 n^\theta$, $\theta$ varying), the system exhibits phase transitions:

• For subcritical $\theta$, spatial homogenization within boxes occurs, and density is frozen.
• At critical scaling ($\theta = 1+\beta$), densities evolve according to a discrete heat equation on box-aggregated densities.
• When both $k, n \to \infty$, the continuous heat equation on the torus emerges (Erhard et al., 5 Dec 2024).

Random Matrices and Universality Classes

Random matrix processes, such as the eigenvalue point processes of products of random matrices or non-Hermitian planar ensembles, exhibit process-level scaling limits dependent on coupling strengths, edge/bulk regimes, and singular potential perturbations. Notable structures include:

• Determinantal or Pfaffian processes with explicit correlation kernels (e.g., Meijer $G$-kernel, Mittag-Leffler functions, fractional differential equation–driven kernels).
• Universality: changing model details (e.g., coupling) may interpolate between fundamentally different scaling limits, but preserve universal behavior in appropriate regimes (1711.01873, Akemann et al., 2021).

Stochastic Processes with Fast Mean Reversion

For Itô diffusions with rapidly mean-reverting components, pathwise averages converge to deterministic integrals with respect to the invariant measure of the fast variable (an averaging principle). Specifically, for $$dX^\varepsilon_t = [b_t - c_t L_t/\varepsilon\, g(M_tX^\varepsilon_t/\varepsilon)]\, dt + \sqrt{c_t}\, dW_t$$,

$$\int_0^t H_s\, f\Bigl(\frac{K_sX^\varepsilon_s}{\varepsilon}\Bigr) ds \longrightarrow \int_0^t H_s\, \bar{f}(K_s, L_s, M_s)\, ds$$

with $\bar{f}(\cdot)$ obtained by integrating $f$ against the invariant density of the frozen fast process (Cayé et al., 2017).

3. Mathematical Structures and Operator Analysis

Scaling limit results typically require the identification of the limiting operator (generator) through asymptotic analysis. Methodologies include:

• Uniform convergence of infinitesimal generators (via core functions) (Möhle et al., 2021, Möhle et al., 2022).
• Characterization of the domain and boundary conditions (e.g., Feller property for contour processes of random trees (Cloez et al., 2018)).
• Introduction of harmonic or scale functions to compute extinction probabilities and describe Doob $h$-transforms for conditioned processes.
• Recursive or functional equations for fixed points and moment-generating functions in processes with infinite memory (Yao, 16 May 2025).

Variational and martingale techniques, tightness (compactness) arguments, and coupling constructions are also standard tools.

4. Dimensionality Reduction and Universality Phenomena

Process-level scaling limits exhibit remarkable dimensionality reduction and universality:

• Infinitely or high-dimensionally parameterized objects, such as random measures or large random graphs, often admit a one-parameter or finite-dimensional asymptotic description via functional limit theorems.
• Different microscopic models, when tuned appropriately (parameter scaling, initial conditions, boundary effects), fall into identical universality classes (e.g., Brownian motion, stable Lévy process, Ornstein–Uhlenbeck, Bessel, stochastic heat equation, etc.).
• In random planar maps, after proper scaling of perimeter and volume, the exploration process converges toward universal continuum objects (hull process of the Brownian plane) via stable Lévy processes and Lamperti transformations (Curien et al., 2014).

5. Applications and Broader Implications

Process-level scaling limits underpin the design and analysis of efficient algorithms (e.g., Markov Chain Monte Carlo with PDMP dynamics in high dimensions (Bierkens et al., 2018)), population genetics (coalescent theory (Möhle et al., 2021, Möhle et al., 2022)), network modeling, statistical mechanics (pinning, wetting, scaling of interfaces (Sohier, 2011, Etheridge et al., 2014)), wireless communication (shot-noise fields (Aburayama et al., 2023)), and deep learning (scaling of wide neural networks, Gaussian Process correspondence, NTK convergence (Yang, 2019)).

They also highlight fundamental barriers in scalable computing, such as strong nonlinear performance limits due to data transfer bottlenecks in large neural network architectures, encapsulated by time-aware versions of Amdahl's Law (Végh, 2020).

6. Open Problems and Future Directions

• Extending process-level scaling limits to new classes of models, especially in the presence of strong heterogeneities, disorder, or high-dimensional dependencies.
• Development of quantitative rates of convergence (e.g., Berry–Esseen bounds for functional convergence) and robust universality criteria.
• Deeper exploration of phase transitions (e.g., condensation, freezing, regime changes in particle systems or combinatorial structures) and their scaling characterizations.
• Leveraging operator-theoretic, potential-theoretic, and stochastic calculus methods to analyze more complex (e.g., non-Markovian, dynamic random environment, or interacting particle) systems.
• Systematic investigation of process-level large and moderate deviations in dependent systems (Yao, 16 May 2025).

7. Summary Table: Select Paradigms of Process-Level Scaling Limits

Setting	Scaling	Limiting Process
LPS Queues (0912.5306)	Time $r^2$, space $1/r$	Reflected Brownian motion (via lifting map)
$\Lambda/\Xi$-coalescents (Möhle et al., 2021, Möhle et al., 2022)	log-centering	Ornstein–Uhlenbeck process (w/ jumps)
SSEP with slow bonds (Erhard et al., 5 Dec 2024)	$n^{1+\beta}$	Discrete/continuous heat equation
Random planar maps (Curien et al., 2014)	$n^{2/3}, n^{4/3}$	Stable Lévy processes, Brownian plane hull
PDMP samplers (Bierkens et al., 2018)	$d\to\infty$	Diffusive/OU limits for angular momentum etc.
Extreme random Young tableaux (Borga et al., 2023)	$N^{-1/2}$	Limit surface via solution to algebraic eqn.
Wide NNs (Yang, 2019)	Layer width $\to \infty$	Gaussian processes, deterministic NTK limit

Process-level scaling limits thus provide a rigorous and unifying framework for understanding the collective evolution of random structures and processes in large-scale or critical regimes, revealing universal behaviors and opening pathways for quantitative approximation and computation across stochastic modeling disciplines.