  Analytic and computational methods developed within statistical physics have
found applications in numerous disciplines. In this letter, we use such methods
to solve a long-standing problem in statistical genetics. The problem, posed by
Haldane and Waddington [J.B.S. Haldane and C.H. Waddington, Genetics 16,
357-374 (1931)], concerns so-called recombinant inbred lines (RILs) produced by
repeated inbreeding. Haldane and Waddington derived the probabilities of RILs
when considering 2 and 3 genes but the case of 4 or more genes has remained
elusive. Our solution uses two probabilistic frameworks relatively unknown
outside of physics: Glauber's formula and self-consistent equations of the
Schwinger-Dyson type. Surprisingly, this combination of statistical formalisms
unveils the exact probabilities of RILs for any number of genes. Extensions of
the framework may have applications in population genetics and beyond.
