I have a biased voter on $\mathbb{Z}^d,$ where $d>0$ (I am mostly interested in the cases where $d>1$) with the bias parameter $\lambda$. In other words, let us have a process $X=(X_t)_{t \ge 0}$ on $S=\{0,1\}^{\mathbb{Z}^d}$ such that if $i,j \in \mathbb{Z}^d$ are two neighbouring sites such that $X_t(i)=1, X_t(j)=0,$ then there a voter move occurs at rate $(1/2d)+\lambda$ changes the state at $j$ to $1$ and the opposite voter move which changes the state at $i$ to $0$ occurs at rate $1/2d$. If I start this biased voter process at the initial state $X_0$ where $X_0(0)=1,X_0(i)=0,i \ne 0,$ then what is the probability of survival of type $1$ until time $t$, i.e. the probability that there exists at least one site at time $t$ such that the state of the process at that site is $1$?
I'm pretty sure this has to be quite standard (I guess I could find it in Liggett's books) but I don't have an access to the library this week (I am sort of on holidays) and I cannot find anything by googling. If you can direct me to a free online source where this is treated then I'd be grateful!
Thanks!
I'm not sure how accurate a result you require, but regardless of the lattice, discrete time Markov chain embedded in $|X_t|$ is a biased random walk that gets absorbed at $0$. From this, you can easily compute the limit as $t\to\infty$ of the survival probability. The papers by Bramson and Griffith on the "Williams-Bjerknes tumor growth" model give bounds on the rate of this convergence (and a limiting shape theorem for $X_t$), which I believe is exponential (or stretched exponential) in $t$.