Multitime Walk Field

• Multitime Walk Field is a random surface constructed from a Poisson line process where each line is marked with a step size, linking random geometry, hydrodynamic limits, and stochastic hard rod dynamics.
• It encapsulates both macroscopic Euler-scale behavior and microscopic diffusive fluctuations, demonstrating convergence to classic Gaussian fields under appropriate scaling.
• The framework unifies ballistic transport in hard rod systems with probabilistic representations, offering insights into invariant measures, ideal gas limits, and interacting particle systems.

The multitime walk field is a random surface constructed from the Poisson line process in the plane, where each line segment is associated with a step size or mark. This object serves as a coupling structure between random geometry (Poisson processes), hydrodynamic limits, and the stochastic dynamics of hard rods. The multitime walk field encodes both the macroscopic (Euler/hydrodynamic) and microscopic (diffusive/Gaussian) scaling limits that arise in interacting particle systems, notably generating the multitime (Lévy–Chentsov) field under diffusive rescaling. Its analysis provides a unified framework for understanding ballistic and diffusive transport in one-dimensional hard-rod systems, as well as their convergence to classic Gaussian fields in the scaling limit (Ferrari et al., 7 Nov 2025).

1. Construction via the Poisson Line Process

The starting point is a Poisson point process $X=\{(x_i,v_i)\}$ on the plane $\mathbb{R}_{x}\times\mathbb{R}_{v}$ with intensity measure $\mu(dx,dv) = \rho(x,v)\,dx\,dv$, typically homogeneous with constant $\lambda$. Each point generates a space-time trajectory (a straight line): $$\ell(x_i,v_i) = \{ (x_i+v_i t,\ t):\ t\in\mathbb{R} \}.$$ The set $\{ \ell(x_i, v_i) \}$ forms the Poisson line process in the plane—the ballistic codification essential for multitime random surfaces and for kinetic models of hard rods.

To each line, an independent real-valued mark $r_i$ (drawn from a probability law $\nu$ with mean zero and finite variance $\sigma^2$) is assigned. The lines, together with their marks, form a marked Poisson process on $$\mathbb{R}_x \times \mathbb{R}_v \times \mathbb{R}_r$$ with product intensity $\mu(dx,dv,dr) = \rho(x,v)\,dx\,dv\,\nu(dr)$.

2. Definition and Structure of the Multitime Walk Field

Given the origin $o = (0,0)$ and any spacetime point $b = (x,t)$, the multitime walk field is constructed by summing the marked steps of all lines crossing the segment $[o, b]$, counted with orientation:

• $ob_+$: lines crossing $[o,b]$ with $o$ on their left and $b$ on their right;
• $ob_-$: same with reversed orientation.

For the marked line $(x_i, v_i, r_i)$, the step surface is: $$h_{x_i,v_i,r_i}(b) = r_i \bigl[ \mathbf{1}\{(x_i,v_i,r_i)\in ob_+\} - \mathbf{1}\{(x_i,v_i,r_i)\in ob_-\} \bigr].$$ The multitime walk field at $b$ is: $$W(b) = \sum_{(x_i,v_i,r_i) \in X} h_{x_i,v_i,r_i}(b).$$ The field increment $W(b) - W(a)$ represents the net signed count (weighted by marks) of lines crossing the portion of space-time between $a$ and $b$, generalizing the notion of a walk to multidimensional time.

3. Hydrodynamic Limit: Law of Large Numbers

The hydrodynamic regime considers the field under intensity scaling $\phi^{-1}\mu$ with $\phi \to \infty$. In this limit, by the law of large numbers for Poisson processes: $$W^\phi(b) \to H_\mu(b) = \int h_{x,v,r}(b) \,\mu(dx,dv,dr)$$ almost surely. This deterministic limit corresponds to the macroscopic Euler equations for hard rod flows, yielding rod density and current in the hydrodynamic limit. In this regime, fluctuations vanish and only the averaged (mean) field persists.

In the context of hard rods, this scaling recovers the classical Euler-scale hydrodynamics and provides the effective velocity formula for the particle current, as derived in the Dobrushin–Boldrighini–Suhov theory.

4. Diffusive Fluctuations: Central Limit and Gaussian Fields

The diffusive scaling (central limit regime) investigates the fluctuation field: $$\eta^\phi(b) = \phi^{-1/2} (W^\phi(b) - \mathbb{E}[W^\phi(b)]).$$ As $\phi \to \infty$, $\eta^\phi$ converges to a centered Gaussian field $\eta(b)$ with covariance: $$\mathrm{Cov}(\eta(a),\eta(b)) = \mu_2(oa \cap ob) = \frac{1}{2}[\mu_2(oa) + \mu_2(ob) - \mu_2(ab)],$$ where $\mu_2(dx,dv,dr) = r^2 \mu(dx,dv,dr)$. For spatially homogeneous and stationary data ($\mu_2(ab) = \sigma^2|b-a|$), this yields the classical Lévy–Chentsov field covariance: $$\mathrm{Cov}(\eta(x,t), \eta(x', t')) = \sigma^2 \min\{x, x'\}\min\{t, t'\},$$ so the diffusive limit of the multitime walk field is the multitime Brownian motion (Lévy–Chentsov field).

This connection establishes that in appropriate scaling, the macroscopic fluctuations of the associated hard-rod system are Gaussian, and the field serves as a universal scaling limit for surface fluctuations in a broad class of non-equilibrium one-dimensional models.

5. One-Dimensional Cuts and Random Walk Projections

Taking a section (or "cut") through the multitime walk field along any straight spacetime path—for example, $x = \bar{x}$—yields a one-dimensional process: $$t \mapsto Z_{\bar x}(t) = W(\bar x, t) - W(\bar x, 0),$$ which is a nonhomogeneous continuous-time random walk whose jump rates and structure are determined by the marks and intersection patterns of the Poisson lines. The nonhomogeneity arises from the random spatial and velocity marginal distributions.

Under diffusive rescaling $\bar{x} \mapsto \phi^{-1/2}\bar{x}$, $t \mapsto \phi^{-1}t$, a central limit theorem holds: $Z_{\bar x}(t)$ converges to standard Brownian motion. Cuts in other directions (e.g., along rays with constant $x + vt$) similarly generate continuous-time random walks that converge to Brownian motion under appropriate scaling.

A cross-sectional interpretation emerges: transversal cuts correspond, in the ideal gas/hard-rod framework, to the evolution of single tagged particles, providing a probabilistic bridge between microscopic walks and macroscopic diffusive behavior.

6. Hard Rod System and Dynamical Interpretation

Interpreting the marked Poisson lines as hard rods, each marked by initial position $x_i$, velocity $v_i$, and length $r_i$, one constructs the hard-rod process:

• Each rod travels at constant speed until it collides with another, at which point the labels are swapped.
• The dynamical state at time $t$ is encoded by: $y_i(0) = x_i + W(x_i, 0),$ so that the time-evolved configuration is: $$U_t\{y_i(0), v_i, r_i\} = \{ y_i(0) + v_i t + W(y_i(0) + v_i t, t),\ v_i, r_i\}.$$ The same surface $W$ thus encodes both the non-interacting (ideal gas) and interacting (hard-rod) dynamics. When rod lengths vanish, one recovers ideal gas (no interaction). When rod lengths are positive, the full interacting hard-rod (ballistic plus collisions) process is recovered.

The law of large numbers and central limit theorem for the surface $W$ then yield—respectively—the Euler-scale hydrodynamic limit and diffusive Gaussian fluctuations for the empirical density and current of hard rods. This provides a direct demonstration that the macroscopic transport of one-dimensional hard rods arises from the underlying multitime random surface, and that fluctuations across these systems are universally Gaussian in the scaling limit.

7. Relation to Invariant Measures and Ideal Gas

In the limiting case where all rod lengths are set to zero, the hard-rod dynamics reduces to that of the ideal gas: rods never collide, and their velocities remain fixed forever. The invariant measure for the ideal gas arises as a particular case of the multitime walk field construction with rod size distribution degenerated at zero. The structure of invariant measures for the hard-rod process is determined by the interplay between the marks (rod lengths), velocity distribution, and the underlying Poissonian geometry. The transition from ideal gas to hard-rod regimes thus interpolates between non-interacting and interacting particle systems within a unified Poisson field framework.

The multitime walk field therefore constitutes an exact probabilistic representation for both microscopic stochastic processes and their macroscopic scaling limits in low-dimensional interacting systems, linking random geometry, kinetic theory, and stochastic hydrodynamics (Ferrari et al., 7 Nov 2025).