Determinantal point processes on complex manifolds: Construction and limit theorems
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916045539770368 |
|---|---|
| author | Lemoine, Thibaut |
| author_facet | Lemoine, Thibaut |
| contents | We develop a coordinate-free probabilistic framework for determinantal point processes associated with Bergman kernels on compact complex manifolds. The basic issue is that Bergman kernels are naturally line-bundle-valued: $B_k(x,y)\in\operatorname{Hom}(L_y^k,L_x^k)$. Hence the usual determinantal formula for correlation functions is not literally a scalar determinant unless one first gives it an intrinsic meaning. We rigorously define this determinant and prove that every finite-dimensional Hilbert space of sections of a Hermitian line bundle gives rise to a genuine finite-rank projection determinantal point process on the base manifold. We then isolate a collection of finite-dimensional transfer principles showing how diagonal asymptotics, near-diagonal asymptotics, Schur complements, Toeplitz trace expansions and determinant asymptotics are converted into probabilistic statements. Specializing to $H^0(M,L^k)$, this gives the Bergman ensemble as the geometric analogue of an orthogonal polynomial ensemble, and some of the transfer principles allow us to recover previously known results of Berman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06955 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Determinantal point processes on complex manifolds: Construction and limit theorems Lemoine, Thibaut Complex Variables Mathematical Physics Probability 32Q10, 60B20, 60D05, 60F05, 60F10 We develop a coordinate-free probabilistic framework for determinantal point processes associated with Bergman kernels on compact complex manifolds. The basic issue is that Bergman kernels are naturally line-bundle-valued: $B_k(x,y)\in\operatorname{Hom}(L_y^k,L_x^k)$. Hence the usual determinantal formula for correlation functions is not literally a scalar determinant unless one first gives it an intrinsic meaning. We rigorously define this determinant and prove that every finite-dimensional Hilbert space of sections of a Hermitian line bundle gives rise to a genuine finite-rank projection determinantal point process on the base manifold. We then isolate a collection of finite-dimensional transfer principles showing how diagonal asymptotics, near-diagonal asymptotics, Schur complements, Toeplitz trace expansions and determinant asymptotics are converted into probabilistic statements. Specializing to $H^0(M,L^k)$, this gives the Bergman ensemble as the geometric analogue of an orthogonal polynomial ensemble, and some of the transfer principles allow us to recover previously known results of Berman. |
| title | Determinantal point processes on complex manifolds: Construction and limit theorems |
| topic | Complex Variables Mathematical Physics Probability 32Q10, 60B20, 60D05, 60F05, 60F10 |
| url | https://arxiv.org/abs/2211.06955 |