How about applications of discrete complex analysis to statistical physics? There was a surge of work this past decade on the subject, such as proofs of conformal invariance of 2-D models (Ising, Potts, Spinglass, O(n),etc.). Before, there were mainly unrigorous physics arguments to prove the various facts involved, such as the value of the Honeycomb Constant. The machinery of SLE and discrete complex analysis has been extremely insightful in the proofs involved. Much of the methodology is based on the foundational work done by Onsager and Baxter decades before.
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