Setup: Two particles are independently performing a discrete random walk on the set of non-negative integers as follows. They are at integer 0 at time t=0. Suppose first particle is at integer n(t) at time t. Then at time t+1, it goes to n(t)+1 with probability 'p' and stays at n(t) with probability 1-p. The other particle follows the same rule independently. They perform this random walk for 'm' steps.
Question: For a given real number K, I am interested in the expectation value of $e^{Kn}$, where $n$ is the number of times both particles are at the same non-zero integer. Intuitively, $n$ can be viewed as the number of times the particles collide after leaving their starting position '0'. Any non-trivial upper bound as a function of K, p, m will be appreciated.