Universal features of exit probability in opinion dynamics models with domain size dependent dynamics

We study the exit probability for several binary opinion dynamics models in one dimension in which the opinion state (represented by $\pm 1$) of an agent is determined by dynamical rules dependent on the size of its neighbouring domains. In all these models, we find the exit probability behaves like a step function in the thermodynamic limit. In a finite system of size $L$, the exit probability $E(x)$ as a function of the initial fraction $x$ of one type of opinion is given by $E(x) = f[(x-x_c)L^{1/\nu}]$ with a universal value of $\nu = 2.5 \pm 0.03$. The form of the scaling function is also universal: $f(y) = [\tanh(\lambda y +c) +1]/2$, where $\lambda$ is found to be dependent on the particular dynamics. The variation of $\lambda$ against the parameters of the models is studied. $c$ is non-zero only when the dynamical rule distinguishes between $\pm 1$ states; comparison with theoretical estimates in this case shows very good agreement.