Speaker
Description
We study the \emph{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-$\tfrac12$ XX quantum chain, which in turn admits a free-fermion representation. This mapping yields closed, finite-size expressions for the time-dependent density and current in terms of modified lattice Bessel functions.
At macroscopic scales we conjecture that the SDEP displays \emph{ballistic}, non-local hydrodynamics governed by the continuity equation
$$
\partial_t \rho+\partial_x j[\rho]=0,\qquad
j[\rho](x)=\sin\!\bigl(\pi\rho(x)\bigr)\,\sinh\!\bigl(\pi\mathcal{H}\rho(x)\bigr),
$$
where (\mathcal{H}) is the periodic 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 and implies ballistic scaling (z=1).
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.
