I was reading Redner's A Kinetic View on Statistical Physics and in the chapter regarding The Voter Model, on page 239, I'm not sure what the first few paragraphs are referring to.
As I understand it, the main thesis of the chapter is that considering the voter model in an arbitrary dimension with a finite population of voters, consensus is achieved in a time that depends solely on system size, spatial dimension and the number of opinion states considered (the chapter goes on to consider only two: 1 and -1).
Yet page 239 seems to reach the conclusion of that, if the initial condition imposed is that we have a single voter at the origin with opinion +1 in a sea of undecided voters, the final state for all voters is to be equiprobably at +1 or at -1. I think that this contradicts the aforementioned thesis.
Let's introduce the definition for the average opinion of each voter:
\begin{equation}\label{OpinionMedia} S(\textbf{x},t) = \sum_{\textbf{x}}s(\textbf{x},t) P(\textbf{s}) \end{equation}
where $\textbf{x}$ is the lattice site, $t$ is time, $s(\textbf{x},t)$ is the opinion state of the voter at site $\textbf{x}$ at time $t$ and $P(\textbf{s})$ is a functional that represents the probability of arriving at configuration $s$, whose evolution is given by:
\begin{equation} \frac{d}{dt}P(\textbf{s}) = -\sum_{\textbf{x}} w_\textbf{x}(s) P(\textbf{s}) + \sum_{\textbf{x}} w_\textbf{x}(s^\textbf{x}) P(s^\textbf{x}) \end{equation}
where $w_\textbf{x}(s)$ is a functional that represents the rate of change of the opinion state of each voter, and is found to be given by the expression:
\begin{equation} w_\textbf{x}(s) = \frac{1}{2}\left[1-\frac{s(\textbf{x},t) \sum_{\textbf{y} \in I_\textbf{x}}s(\textbf{y},t)}{z}\right] \end{equation}
where $z$ is the coordination number of the lattice and $I_\textbf{x}$ are the neighbors of $x$.
Working from these definitions, in the chapter we arrive at the following expression for the evolution of the average opinion:
\begin{equation} \frac{d}{dt}S(\textbf{x},t) = -S(\textbf{x},t) + \frac{1}{z} \sum_{\textbf{y} \in I_\textbf{x}}S(\textbf{y},t) \end{equation}
So, in page 239, it starts assuming that the initial condition that we impose on the system is $S(\textbf{x},t=0) = \delta_{x,0}$. It says that this condition means that we have a single voter at the origin that is at opinion state +1 in a sea of undecided voters. I don't understand why this is, could anyone help me gain an intuition of it? For starters, I understand $S(\textbf{x},t)$ to be an expected value. However, $\delta_{x,0}$ is a distribution. It is the distribution that results from the degenerate random variable. So, how can we equal these two notions? Furthermore, if a voter is undecided, that is, it has opinion +1 and -1 equiprobably, it makes sense to me that $S(\textbf{x},t) = \frac{1}{2}*1 + \frac{1}{2}*(-1) = 0$. But, if a voter is at state +1, we'd have $S(\textbf{x},t) = 1*1 + 0*(-1) = 1$, and $\delta_{x,0}(0) = +\infty$ (this is not too rigorous, but it is the usual intuition).
From that point on, to be honest, I struggle to understand what Redner does to reach the conclusion of that asymptotically a single voter relaxes to the average undecided opinion of the rest of the population, and most importantly how is this not in contradiction with what we've stated before. I attach the fragment of the text:
Thank you for your time, I hope that I've included enough information.