Alfonso Garmendia, Leonid Ryvkin
Enrico Reiß (Universität Potsdam)
In mean field theories of statistical mechanics where N particles interact, macroscopic observables typically take values in a scaled lattice with lattice spacing 1/N. An easy but typical example is the magnetization in the Curie-Weiss model (which is one-dimensional). Time evolution in these models can be described by a reversible Markov chain in discrete time, with transition matrix P. Then 1 - P is analog to a self-adjoint infinitesimal generator, or in physical language to the Hamilton operator of the system.
After h-transform with respect to the reversible measure one has transformed to a space with counting measure. Hilbert space theory on this space can be written down by use of Fourier transform, which allows to realize these Hamilton operators as special pseudodifferential operators where each operator arises from a symbol on phase space by quantization.
We show that in the limit "N to infty" (corresponding to lattice spacing tending to zero) the sharp Weyl estimates for the number of eigenvalues inside a compact interval extend from the continuous setting to the discrete setting (under appropriate assumptions). In particular, the leading term is a phase space volume.