We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.
"Large deviation rate functions for the partition function in a log-gamma distributed random potential." Ann. Probab. 41 (6) 4248 - 4286, November 2013. https://doi.org/10.1214/12-AOP768