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Abstract |
We identify the persistence probability for the spin located at the origin of a half-space magnetized Glauber-Ising chain as a Fredholm Pfaffian gap probability generating function with a sech-kernel. This is then recast as a tau-function for a certain Painlev\'e VI transcendent --- a sort of exact Kramers' formula for the associated explicitely time-dependent Hamiltonian --- where the persistence exponent emerges as an asymptotic decay rate. By a known yet remarkable correspondence that relates Painlev\'e equations to Bonnet surfaces, the persistence probability has also a geometric meaning à la Gauss-Bonnet in terms of the intrinsic curvature of the underlying surface. Since the same sech-kernel with an underlying Pfaffian structure shows up in a variety of Gaussian first-passage problems, our Painlev\'e VI characterization appears as a universal probability distribution akin to the famous Painlev\'e II Tracy-Widom laws. Its tail behavior in the magnetization-symmetric 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 super-universal persistence exponent for the coarsening of a non-conserved scalar order parameter in two space dimensions. |
