  Motivated by the phenomenon of duality for interacting particle systems we
introduce two classes of Pfaffian kernels describing a number of Pfaffian point
processes in the `bulk' and at the `edge'. Using the probabilistic method due
to Mark Kac, we prove two Szeg\H{o}-type asymptotic expansion theorems for the
corresponding Fredholm Pfaffians. The idea of the proof is to introduce an
effective random walk with transition density determined by the Pfaffian
kernel, express the logarithm of the Fredholm Pfaffian through expectations
with respect to the random walk, and analyse the expectations using general
results on random walks. We demonstrate the utility of the theorems by
calculating asymptotics for the empty interval and non-crossing probabilities
for a number of examples of Pfaffian point processes: coalescing/annihilating
Brownian motions, massive coalescing Brownian motions, real zeros of Gaussian
power series and Kac polynomials, and real eigenvalues for the real Ginibre
ensemble.
