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Proof of the ASM-DPP conjecture

Mathematical Physics & Probability

Speaker: Philippe Di Francesco, Institut de Physique Theorique,CEA Saclay
Location: 1147 MSB
Start time: Tue, Nov 23 2010, 4:10PM

We prove a 27-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 6Vertex model for ASMs and Rhombus tilings 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)