Emergent hydrodynamics in an exclusion process with long-range interactions

We study the symmetric Dyson exclusion process (SDEP)—a lattice gas with exclusion and long-range, Coulomb–type interactions that emerge both as the maximal-activity limit of the symmetric exclusion process and as a discrete version of Dyson’s Brownian motion on the unitary group. Exploiting an exact ground-state (Doob) transform, we map the stochastic generator of the SDEP onto the spin- \tfrac12 1 2 XX quantum chain, which in turn admits a free-fermion representation. At macroscopic scales we conjecture that the SDEP displays ballistic (Eulerian) scaling non-local hydrodynamics governed by the equation \partial_t \rho+\partial_x j[\rho]=0 ∂ t ρ + ∂ x j [ ρ ] = 0 , j[\rho](x,t)=\frac{1}{\pi}\sin\bigl(\pi\rho(x,t)\bigr)\sinh\bigl(\pi\mathcal{H}\rho(x,t)\bigr) j [ ρ ] ( x , t ) = 1 π sin ⁡ ( π ρ ( x , t ) ) sinh ⁡ ( π ℋ ρ ( x , t ) ) , where \mathcal{H} ℋ is the Hilbert transform, making the current a genuinely non-local functional of the density. This non-local one-field description is equivalent to a local two-field “complex Hopf” system for finite particle density. Closed evolution formulas allow us to solve the melting of single- and double-block initial states, producing limit shapes and arctic curves that agree with large-scale Monte-Carlo simulations. The model thus offers a tractable example of emergent non-local hydrodynamics driven by long-range interactions.