Equality of outcomes in two Poisson events I have a Poisson process with a fixed (large) $\lambda$.  If I run the process twice, what is the probability that the two runs have the same outcome?
That is, how can I approximate
$$f(\lambda)=e^{-2\lambda}\sum_{k=0}^\infty\frac{\lambda^{2k}}{k!^2}$$
for $\lambda\gg1$?  If there's a simple expression about $+\infty$ that would be best, but I'm open to whatever can be suggested.

Another way, similar to Michael, more quick and informal, to get an asymptotic:
We have two independent poisson variables $X$ and $Y$, with parameter $\lambda \gg 1$ and we want to estimate $P(Z=0)$ where $Z=X-Y$. But using the normal approximation, we have $X \approx  N(\lambda,\lambda)$, and so $Y$; and hence $Z \approx N(0,2\lambda)$  ($N$ here denotes the Normal distribution). 
We are approximating here a discrete variable by a continuous one, and so we can further approximate:
$$ P(Z=0) \approx \int_{-1/2}^{1/2} f_z(Z) dZ \approx f_z(0) = N(0,2\lambda)|_{z=0} = \frac{1}{\sqrt{4 \pi \lambda}} $$
