Local scaling asymptotics for the Gutzwiller trace formula in Berezin-T\"oplitz quantization

Under certain hypothesis on the underlying classical Hamiltonian flow, we produce local scaling asymptotics in the semiclassical regime for a Berezin-T\"oplitz version of the Gutzwiller trace formula on a quantizable compact K\"ahler manifold, in the spirit of the near-diagonal scaling asymptotics of Szeg\"o and T\"oplitz kernels. More precisely, we consider an analogue of the \lq Gutzwiller-T\"oplitz kernel\rq\, previously introduced in this setting by Borthwick, Paul and Uribe, and study how it asymptotically concentrates along the appropriate classical loci defined by the dynamics, with an explicit description of the exponential decay along normal directions. These local scaling asymptotics probe into the concentration behavior of the eigenfunctions of the quantized Hamiltonian flow. When globally integrated, they yield the analogue of the Gutzwiller trace formula.