A zero-range process with a dynamic edge-gating environment: exact reversible measure, load-coupled critical density, and metastability on dumbbell graphs

We study a zero-range process (ZRP) on a finite connected graph G_0 = (V, E) coupled to a dynamic binary on-site environment. Each vertex v carries an activity I_v in {0, 1}; an edge {i, j} is open iff I_i I_j = 1, so the environment gates the active subgraph. In turn, I_v flips with rates that depend on the local particle number eta_v. The coupling is two-way: particles drive the environment and the environment selects the active subgraph. The environment is neither a second conserved particle species (as in two-species ZRP) nor quenched disorder; it is a non-conserved edge-gating field with feedback to the particles.

Result 1. For arbitrary positive rate functions alpha, beta and any ZRP jump rate u, the joint generator on the state space Z_+^V x {0,1}^V is reversible with respect to the explicit product measure

Pi(eta, I) ~ prod_v p(eta_v) (k_v)^{eta_v} [ 1{I_v=1} + (alpha/beta)(eta_v) 1{I_v=0} ]

on the canonical sector, where p is the standard ZRP weight associated with u and k_v is the degree of v. The measure factorises as an eta-marginal times a state-dependent Bernoulli field for I.

Result 2. For affine coupling alpha(n) = A, beta(n) = 1 + n and jump rate u(n) = 1 + b/n with b > 3, the eta-marginal admits a closed-form effective critical density rho_c(A) = [ A(b-3) + (b+1) ] / [ (b-3) (A(b-2) + 1) ], monotone decreasing in A and recovering the Evans value 1/(b-2) as A tends to infinity.

Result 3 (in progress). On the dumbbell graph, the Z_2 graph automorphism is preserved by the equilibrium measure but generically broken in the supercritical phase. Using the Beltran-Landim capacity method, we obtain the leading order of the inter-clique transition rate of the condensate as a function of (m, N, L, A).