We prove a 28-years old conjecture by Mills-Robbins-Rumsey (1983) relating some refined enumerations of Alternating Sign Matrices (ASM) and Descending Plane Partitions (DPP). These are performed by reformulating the enumeration problems in terms of statistical models, namely the 6-vertex model for ASMs and Rhombus tilings/Dimers or Lattice Paths for DPPs. The conjecture then boils down to a determinant identity, which is proved by use of generating function techniques. Remarkably, the main player is the transfer matrix for discrete 1+1-dimensional Lorentzian quantum gravity, which generates random Lorentzian triangulations of the two-dimensional space-time. (This is joint work with Roger Behrend and Paul Zinn-Justin).

IPhT Saclay