  The purpose of this article is to develop a theory behind the occurrence of
"path-integral" kernels in the study of extended determinantal point processes
and non-intersecting line ensembles. Our first result shows how determinants
involving such kernels arise naturally in studying ratios of partition
functions and expectations of multiplicative functionals for ensembles of
non-intersecting paths on weighted graphs. Our second result shows how Fredholm
determinants with extended kernels (as arise in the study of extended
determinantal point processes such as the Airy_2 process) are equal to Fredholm
determinants with path-integral kernels. We also show how the second result
applies to a number of examples including the stationary (GUE) Dyson Brownian
motion, the Airy_2 process, the Pearcey process, the Airy_1 and Airy_{2->1}
processes, and Markov processes on partitions related to the z-measures.
