Current fluctuations for the boundary-driven zero-range process on graphs: microscopic versus macroscopic approach and a theory of non-reversible resistor-like networks

We compute the joint large deviation rate functional in the limit of large time for the current flowing through the edges of a finite graph on which a boundary-driven system of stochastic particles evolves with zero-range dynamics.This generalizes one-dimensional results previously obtained with different approaches; our alternative techniques illuminate various connections and complementary perspectives. In particular, we here use a variational approach to derive the rate functional by contraction from a level 2.5 large deviation rate functional. We perform an exact minimization and finally obtain the rate functional as a variational problem involving a superposition of cost functions for each edge. The contributions from different edges are not independent since they are related by the values of a potential function on the nodes of the graph. The rate functional on the graph is a microscopic version of the continuous rate functional predicted by the macroscopic fluctuation theory, and we indeed show a convergence in the scaling limit. If we split the graph into two connected regions by a cutset and are interested just in the current flowing through the cutset, we find that the result is the same as that of an effective system composed of only one effective edge (as happens at macroscopic level and is expected also for other models). The characteristics of this effective edge are related to the ``capacities’’ of the graph and can be obtained by a reduction using elementary transformations as in electrical networks; specifically, we treat components in parallel, in series, and in $N$-star configurations (reduced to effective complete $N$-graphs). Our reduction procedure is directly related to the reduction to the trace process and, since the dynamics is in general not reversible, it is also closely connected to the theory of non-reversible electrical networks.

💡 Research Summary

The paper investigates the large‑deviation properties of particle currents in a boundary‑driven zero‑range process defined on a finite directed graph. Particles hop between internal vertices with rates that depend on the local occupation number through a super‑linear function g, and they are injected or removed at boundary (“ghost”) vertices with rates determined by λ and edge weights c. Under the super‑linear growth assumption, condensation is avoided and the process admits a unique product‑type invariant measure.

The authors start from the level 2.5 large‑deviation principle, which provides a joint rate functional I(μ,Q) for the empirical measure μ_T and empirical flow Q_T of the underlying Markov chain. By exploiting the super‑linear condition they ensure that the empirical flow lives in the space L¹⁺ of summable, divergence‑free flows, making the level 2.5 LDP applicable despite the infinite state space.

A key contribution is the contraction of this level 2.5 functional onto the space of edge currents j_{xy} and node potentials ψ_x. Solving the resulting variational problem exactly yields a cost for each directed edge of the form
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