On the probability of the existence of path from one corner to its opposite

Question

Consider a $n\times n$ grid, whose nodes are randomly colored black and white (with probability $p$ and $1-p$ respectively). Let $A$ be the event that there exists a path of all black nodes connecting the top-left corner and the bottom-right corner, and $B$ be the event that there exists a path of all black nodes connecting the bottom-left corner and the top-right corner.

Now please prove that $P(B|A)\ge P(B) $

It can be understood intuitively that $A$ and $B$ are positively correlated, however, there is possibly no closed form of these probabilities (see this problem). Therefore pure calculation might not be able to solve this problem.

Answer 1

Your question is an elementary consequence to the fact that the site percolation model (as you defined it) satisfies the Fortuin-Kasteleyn-Ginibre inequality for increasing events.

In your model, say that two colorings $\sigma$ and $\tau$ satisfy $$\sigma \leq \tau$$ if and only if $$\sigma(v) \leq \tau(v) \text{ for all gridpoints }v\,,$$ where we say that 'white < black'. This defines a partial order on colorings. Call an event $A$ increasing if $\sigma \in A$ and $\sigma \leq \tau$ implies $\tau \in A$. The events $A$ and $B$ of diagonal black crossings you described are increasing events w.r.t. this partial order.

The FKG-inequality states that

$$P(A\cap B) \geq P(A) P(B) $$ for increasing events $A$ and $B$, or equivalently $$ P(B\mid A) \geq P(B)\,.$$

To see that this is true, consider all colorings in $\sigma \in A$ and define the following Markov chain:

• For a given $\tau \in A$, choose a vertex $v$ randomly
• Recolor $\tau(v)$ white. If this new configuration is in $A$, accept it with probability $1-p$, otherwise recolor it black.

The stationary distribution of this Markov chain is exactly $P(\cdot \mid A)$. Call this Markov chain $(\tau^t)$, with $tau^0$ the all-black coloring. Now, define the chain $\sigma^t$ via the process

• For a given $\sigma$, choose a vertex $v$ randomly
• Recolor $\sigma(v)$ white with probability $1-p$, otherwise black.

with $\sigma^0$ the al-white coloring. The stationary distribution of this Markov chain obviously is $P(\cdot)$. Now, we have $\sigma^0\leq \tau^0$, and the Markov chains can be coupled to ensure $\sigma^t \leq \tau^t$ for all $t\geq 0$ (simply choose the same vertex in Step 1, and base the choice in Step 2 on the same random number generated uniformly in $[0,1]$). As a consequence, we get $$P(B) = \lim_{t\to\infty} P(\sigma^t\in B) \leq \lim_{t\to\infty} P(\tau^t\in B) = P(B\mid A)\,.$$