Partial Bergman kernels and determinantal point processes on Kähler manifolds
Louis Ioos
TL;DR
This work develops full off-diagonal asymptotics for equivariant and partial Bergman kernels on prequantized Kähler manifolds with circle actions and bounded geometry, then translates these kernel expansions into probabilistic statements for the associated determinantal point processes. The authors prove a Law of Large Numbers for linear statistics and a Central Limit Theorem with a variance formula that splits into a bulk $H^1$-norm term on the droplet $\{\mu<0\}$ and a boundary $H^{1/2}$-norm term on the reduced boundary $X_0=\mu^{-1}(0)/S^1$, namely $\lim_{p\to\infty} p^{-(n-1)}\mathrm{Var}[\mathcal N_p[f]] = \frac{1}{4\pi}\int_{\{\mu<0\}} |df|^2\,dv_X + \frac{1}{2}\int_{X_0} \sum_{k\in\mathbb Z} |k| |\hat f_k|^2\,dv_{X_0}$, with a CLT scaling exponent $\alpha=\frac{1}{2n}-\frac{1}{2}$. These results generalize the Ginibre/circular law scenario and extend Berman’s work to nontrivial circle actions and noncompact settings, revealing a deep link between quantization, symplectic reduction, Szegő theory, and fluctuations of DPPs on Kähler manifolds. The approach hinges on full off-diagonal Bergman kernel expansions (via Ma–Marinescu/Dai–Liu–Ma theory) for both equivariant and partial kernels, and their careful use to control linear statistics through Fourier/weight-space decompositions and Euler–Maclaurin type arguments.
Abstract
We compute the full off-diagonal asymptotics of the equivariant and partial Bergman kernels associated with a circle action on a prequantized Kähler manifold with bounded geometry at infinity, then use these results to compute the asymptotics of the linear statistics of the associated determinantal point process as the number of points grows to infinity, showing that its distribution converges to a centered normal variable with variance given by the sum of an $H^1$-norm squared in the bulk and an $H^{1/2}$-norm squared on the boundary of the associated droplet.