Size of connected regions on a randomly-colored infinite chessboard

Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We can consider the entire connected region $W$ of white squares that includes the white square at $(0,0)$; that is, the set of all squares that are reachable from $(0,0)$ via an all-white path of adjacent squares. (Squares are deemed to be adjacent if they share an edge.) The expected size $E_W$ of $W$ is a monotonically-increasing function of $p$ with minimum value 1.

At what value of $p$ does $E_W$ become infinite? (That is, what is $\inf \{p : E_W(p) = \infty \}?)$

(Clearly there is such a value, since for $p$ close to 1, nearly all the squares are white, and nearly all of them are reachable from $(0,0)$.)

I would especially like a reference to a monograph on this topic and topics related to it. (What if the chessboard is partitioned into more than 2 pieces? What if it is replaced with some other graph? What is the probability $p_\infty$ that region $W$ is infinite, as a function of $p$? At what value of $p$ does $p_\infty$ become positive? If $r_p(s)$ is the probability that the region containing $(0,0)$ has size exactly $s$ squares when squares are white with probability $p$, just how is $r_p$ distributed?) Is Roach The theory of random clumping (Methuen, 1968) what I am looking for?

(Please retag this question as appropriate.)

As rightly mentioned in the comments, the needed keyword is "site percolation on square lattice". The expected size of a cluster containing the origin indeed jumps from 0 to infinity when $p$ passes the site percolation theshold for this lattice, $p_c$. In the vicinity of $p=p_c$, $p_\infty=0$ at $p<p_c$, and otherwise $p_\infty\propto(p-p_c)^\beta$ where $\beta=5/36$ is called a critical exponent. For more information about clusters on percolating lattices and critical exponents in percolation theory, including cluster number distribution, check out the lecture notes by Christensen.