Large deviation principles for some random combinatorial structures in population genetics and Brownian motion

Large deviation principles are established for some random combinatorial structures including the Ewens sampling formula and the Pitman sampling formula. A path-level large deviation principle is established for the former on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence topology, and the rate function is the same as for a Poisson process justifying the Poisson process approximation for the Ewens sampling formula at the large deviation level. A large deviation principle for the total number of parts in a partition is obtained for the Pitman formula; here the rate function depends only on one of the two parameters which display the different roles of the two parameters at different scales. In addition to these large deviation results, we also provide an embedding scheme which gives the Pitman sampling formula. A product of this embedding is an intuitive alternate proof of a result of Pitman on the limiting total number of parts.