The Library
Determinantal structure of conditional overlaps in the complex Ginibre ensemble
Tools
Tsareas, Athanasios (2021) Determinantal structure of conditional overlaps in the complex Ginibre ensemble. PhD thesis, University of Warwick.
|
PDF
WRAP_Theses_Tsareas_2021.pdf - Submitted Version - Requires a PDF viewer. Download (1005Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3763665~S15
Abstract
The thesis is split in two parts. First part focuses on conditional overlaps and the contents of our paper. Section 2 presents our main results concerning conditional expectations of overlaps: the determinantal representation for n < ∞, the bulk and the edge scaling limits, exact algebraic asymptotic in the bulk for well separated eigenvalues. Section 3 contains the proofs as presented in our paper [2]: 3.1, 3.2 is the derivation of the determinantal representation for the conditional expectations of diagonal and off diagonal overlaps in terms of bi-orthogonal polynomials in the complex plane; 3.3 a heuristic calculation of the correlation kernels, which shows how the result of rather complicated calculations of the following sections can be easily guessed using the assumption of the extended translational invariance; 3.4 a rigorous evaluation of correlation kernels for n < ∞ in terms of the exponential polynomials; 3.5 - 3.7 the calculation of various scaling limits as n → ∞. The methods used in these proofs are rather classical: the determinantal structure is a consequence of Dyson’s theorem reviewed in [24] and the product structure of the overlap expectations conditioned on all eigenvalues; the computation of the correlation kernel for the diagonal overlaps reduces to the inversion of the tri-diagonal moment matrix using the recursions already encountered in [10], [11] and [32]; the calculation of the kernel for the off-diagonals overlaps uses a relation between diagonal and off-diagonal overlaps established in Lemma 1 and determinantal identities, which we explore more thoroughly in Section 5. More results can be found in both [2] and our follow up paper [1].
In Section 4 we go and explore alternate approaches and proof. We show that some of the tri-diagonal can be computed directly, we present different ways of finding the kernel and the orthogonal polynomials, a different proof of Theorem 1 using contour integrals and a more detailed version of the bulk and edge limits computations of the kernel. These alternative methods were explored at the same time as the methods that eventually were included in the paper. We hope that both sets of methods may prove useful in generalisations of these results.
The second part of the thesis revolves around a parallel research on the conditioning of determinantal and pfaffian point processes. It is general results, which I illustrate through Ginibre, but otherwise unrelated. In Section 5 we use a presumably known, but long forgotten determinantal identity, and use it to compute explicitly the kernel of conditioned determinantal and pfaffian p.p. on occupied points, empty sets or the combination of the two. We explore the discrete determinantal case, the discrete pfaffian case by finding an analogue of said identity, but for pfaffians using Tanner’s identity [21] and finally the continuous determinantal point processes with the assistance of Campbell and Palm measures. Unfortunately the continuous Pfaffian remains for the time being unfinished. A more detailed account of the work can be found in Section 5.
Item Type: | Thesis (PhD) | ||||
---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Random matrices, Pfaffian systems, Determinants | ||||
Official Date: | 2021 | ||||
Dates: |
|
||||
Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Sponsors: | Tribe, Roger ; Zaboronski, Oleg V. | ||||
Extent: | 126 leaves | ||||
Language: | eng |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year