Exact Results for the Symmetric Dyson Exclusion Process

The symmetric Dyson exclusion process (SDEP) is an exclusion process on the lattice with a long-range logarithmic Coulomb-type interaction. It appears in several equivalent forms: as symmetric random walkers conditioned, in the Doob-transform sense, not to collide; as the maximal-activity limit of a conditioned SSEP; and as a ground-state transform of the spin- 1/2 XX chain. In this work, we exploit this latter representation to obtain exact evolution formulas from deterministic initial configurations. The resulting determinantal kernel is expressed through a finite interpolation expression in terms of Lagrange polynomials, leading to explicit density evolution in finite and infinite lattices. For the melting of a densely packed block, we show that the full hierarchy of density moments is governed by a finite-dimensional polynomial algebra, and we identify Catalan numbers in the leading time coefficients. We also derive the Euler-scale density profile and the arctic curve separating frozen and liquid regions, and show that they coincide with conjectured hydrodynamic results obtained in a previous work.