We introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint. The probability of a pair-annihilation event decays as a power-law in the distance between particles, with a tunable exponent. Such long-range processes arise naturally in various classical settings, and increasingly play a role in the study of quantum dynamics: for example, quantum state preparation or error correction processes aim to prepare ordered ground states, which requires removing pointlike excitations in pairs via nonlocal feedback operations conditioned on a global set of measurement outcomes. We analytically and numerically describe features of absorbing phases and phase transitions in this family of classical models as pairwise annihilation is performed at larger and larger distances. Notably, we find that the two canonical absorbing-state universality classes—directed percolation and parity conserving–are endpoints of a new line of universality classes with continuously varying critical exponents. These new classes follow from an interplay of long-range interactions and symmetries: decreasing the long-range exponent effectively weakens the local effects of global symmetry constraints. This competition significantly broadens known nonequilibrium phases and phase transitions.

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• Phase transitions
• Nonequilibrium systems
• Finite-size scaling
• Mean field theory
• Monte Carlo methods