Virtual walks inspired by a mean field kinetic exchange model of opinion dynamics

We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter $p$ which drives an order disorder transition at a critical value $p_c$. The distribution $S(X,t)$ of the displacements $X$ from the origin of the walkers is computed at different times. Below $p_c$, two time scales associated with a crossover behavior in time are detected, which diverge in a power law manner at criticality with different exponent values. $S(X,t)$ also carries the signature of the phase transition as it changes its form at $p_c$. The walks show the features of a biased random walk below $p_c$, and above $p_c$, the walks are like unbiased random walks. The bias vanishes in a power law manner at $p_c$ and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies.