{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.15479/at:ista:17164","date_updated":"2025-01-09T09:49:31Z","year":"2024","publisher":"Institute of Science and Technology Austria","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020"}],"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","article_processing_charge":"No","has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"ddc":["519"],"abstract":[{"lang":"eng","text":"This thesis is structured into two parts. In the first part, we consider the random\r\nvariable X := Tr(f1(W)A1 . . . fk(W)Ak) where W is an N × N Hermitian Wigner matrix, k ∈ N, and we choose (possibly N-dependent) regular functions f1, . . . , fk as well as\r\nbounded deterministic matrices A1, . . . , Ak. In this context, we prove a functional central\r\nlimit theorem on macroscopic and mesoscopic scales, showing that the fluctuations of X\r\naround its expectation are Gaussian and that the limiting covariance structure is given\r\nby a deterministic recursion. We further give explicit error bounds in terms of the scaling\r\nof f1, . . . , fk and the number of traceless matrices among A1, . . . , Ak, thus extending\r\nthe results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length k ≥ 2.\r\nAnalyzing the underlying combinatorics leads to a non-recursive formula for the variance\r\nof X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1 . . . fk+ℓ(W)Ak+ℓ) of similar\r\nbuild. When restricted to polynomials, these formulas reproduce recent results of Male,\r\nMingo, Peché, and Speicher [107], showing that the underlying combinatorics of noncrossing partitions and annular non-crossing permutations continue to stay valid beyond\r\nthe setting of second-order free probability theory. As an application, we consider the\r\nfluctuation of Tr(eitW A1e\r\n−itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t\r\nis large and give an explicit formula for the variance.\r\nThe second part of the thesis collects three smaller projects focusing on different random\r\nmatrix models. In the first project, we show that a class of weakly perturbed Hamiltonians\r\nof the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization.\r\nThat is, the time evolution generated by Hλ relaxes to its ultimate thermal state via an\r\nintermediate prethermal state with a lifetime of order λ\r\n−2\r\n. As the main result, we obtain\r\na general relaxation formula, expressing the perturbed dynamics via the unperturbed\r\ndynamics and the ultimate thermal state. The proof relies on a two-resolvent global law\r\nfor the deformed Wigner matrix Hλ.\r\nThe second project focuses on correlated random matrices, more precisely on a correlated N × N Hermitian random matrix with a polynomially decaying metric correlation\r\nstructure. A trivial a priori bound shows that the operator norm of this model is stochastically dominated by √\r\nN. However, by calculating the trace of the moments of the matrix\r\nand using the summable decay of the cumulants, the norm estimate can be improved to a\r\nbound of order one.\r\nIn the third project, we consider a multiplicative perturbation of the form UA(t) where U\r\nis a unitary random matrix and A = diag(t, 1, ..., 1). This so-called UA model was\r\nfirst introduced by Fyodorov [73] for its applications in scattering theory. We give a\r\ngeneral description of the eigenvalue trajectories obtained by varying the parameter t and\r\nintroduce a flow of deterministic domains that separates the outlier resulting from the\r\nrank-one perturbation from the typical eigenvalues for all sub-critical timescales. The\r\nresults are obtained under generic assumptions on U that hold for various unitary random\r\nmatrices, including the circular unitary ensemble (CUE) in the original formulation of\r\nthe model."}],"type":"dissertation","degree_awarded":"PhD","title":"Central limit theorems for random matrices: From resolvents to free probability","month":"06","status":"public","ec_funded":1,"oa_version":"Published Version","tmp":{"name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)"},"_id":"17164","day":"26","keyword":["Random Matrices","Spectrum","Central Limit Theorem","Resolvent","Free Probability"],"publication_status":"published","page":"206","oa":1,"alternative_title":["ISTA Thesis"],"date_published":"2024-06-26T00:00:00Z","file_date_updated":"2024-06-26T12:44:53Z","citation":{"mla":"Reker, Jana. <i>Central Limit Theorems for Random Matrices: From Resolvents to Free Probability</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:17164\">10.15479/at:ista:17164</a>.","ama":"Reker J. Central limit theorems for random matrices: From resolvents to free probability. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:17164\">10.15479/at:ista:17164</a>","apa":"Reker, J. (2024). <i>Central limit theorems for random matrices: From resolvents to free probability</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:17164\">https://doi.org/10.15479/at:ista:17164</a>","short":"J. Reker, Central Limit Theorems for Random Matrices: From Resolvents to Free Probability, Institute of Science and Technology Austria, 2024.","ista":"Reker J. 2024. Central limit theorems for random matrices: From resolvents to free probability. Institute of Science and Technology Austria.","ieee":"J. Reker, “Central limit theorems for random matrices: From resolvents to free probability,” Institute of Science and Technology Austria, 2024.","chicago":"Reker, Jana. “Central Limit Theorems for Random Matrices: From Resolvents to Free Probability.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:17164\">https://doi.org/10.15479/at:ista:17164</a>."},"corr_author":"1","date_created":"2024-06-24T11:23:29Z","publication_identifier":{"issn":["2663-337X"]},"supervisor":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"}],"file":[{"relation":"main_file","access_level":"open_access","content_type":"application/pdf","creator":"jreker","file_size":2783027,"file_name":"ISTA_Thesis_JReker.pdf","date_created":"2024-06-26T12:39:36Z","file_id":"17176","checksum":"fb16d86e1f2753dc3a9e14d2bdfd84cd","date_updated":"2024-06-26T12:44:53Z"},{"file_id":"17177","checksum":"cb1e54009d47c1dcf5b866c4566fa27f","date_updated":"2024-06-26T12:44:53Z","file_name":"ISTA_Thesis_JReker_SourceFiles.zip","file_size":3054878,"creator":"jreker","date_created":"2024-06-26T12:39:42Z","relation":"source_file","access_level":"closed","content_type":"application/zip"}],"author":[{"full_name":"Reker, Jana","first_name":"Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","last_name":"Reker"}],"language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"part_of_dissertation","id":"17174","status":"public"},{"id":"17173","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"11135"},{"status":"public","id":"17154","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"17047","status":"public"}]}}