Status  Confirmed 
Seminar Series  IPHTPHM 
Subjects  mathph 
Date  Monday 1 October 2018 
Time  11:00 
Institute  IPHT 
Seminar Room  Salle Claude Itzykson, Bât. 774 
Speaker's Last Name  Ivan Dornic 
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Speaker's Institution  SPEC, CEA/Saclay 
Title  A tale of Pfaffian persistence tails told by a Painlevé VI transcendent 
Abstract  We identify theÂ persistence probability for the spin located at the origin of a halfspace magnetized GlauberIsing chain as a Fredholm Pfaffian gap probability generating function with a sechkernel. This is then recast asÂ a taufunction for a certain Painlevé VI transcendent  a sort of exact Kramers' formulaÂ for the associated explicitelyÂ timedependent Hamiltonian Â where the persistence exponent emerges as an asymptotic decay rate. By aÂ known yet remarkable correspondence that relates Painlevé equations to Bonnet surfaces, the persistence probability has also a geometric meaning Ã la GaussBonnet in terms of the intrinsic curvature of the underlying surface. Since the sameÂ sechkernel with an underlying Pfaffian structure shows up in a variety of Gaussian firstpassage problems, our Painlevé VI characterization appears as aÂ universal probability distribution akin to the famous Painlevé II TracyWidom laws. Its tail behaviorÂ in the magnetizationsymmetric case allows in particular to recover the exact value 3/16 for the persistence exponent of a 2d diffusing random field, as found very recently by Poplavskyi and Schehr (arXiv:1806.11275). Due to its topological origin, this value should constitute the superuniversal persistence exponent for the coarsening of a nonconserved scalar order parameter in two space dimensions. 
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