Exit probability in inflow dynamics: nonuniversality induced by range, asymmetry and fluctuation

Probing deeper into the existing issues regarding the exit probability (EP) in one dimensional dynamical models, we consider several models where the states are represented by Ising spins and the information flows inwards. At zero temperature, these systems evolve to either of two absorbing states. The exit probability $E(x)$, which is the probability that the system ends up with all spins up starting with $x$ fraction of up spins is found to have the general form $E(x) = x^α/\left[x^α+ (1-x)^α\right]$. The exit probability exponent $α$ strongly depends on $r$, the range of interaction, the symmetry of the model and the induced fluctuation. Even in a nearest neighbour model, nonlinear form of EP can be obtained by controlling the fluctuations and for the same range, different models give different results for $α$. Non-universal behaviour of the exit probability is thus clearly established and the results are compared to existing studies in models with outflow dynamics to distinguish the two dynamical scenarios.

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