Combinatorics of Classical Unitary Invariant Ensembles and Integrable Systems

The first part of this thesis is devoted to the combinatorics, geometry, and effective computation of correlators of unitary invariant ensembles of random hermitian matrices with classical potentials. The main results are the subject of the publications [7, 8] with my supervisors T.~Grava and G.~Ruzza, and are summarized as follows. We provide generating functions for correlators of general Hermitian matrix models; formulae of this sort have already appeared in the literature [1, 5], we rederive them here with different methods which lend themselves to further generalizations. Such formulae are not recursive in the genus and hence particularly effective. Moreover, these formulae express the correlators of classical unitary ensembles as linear combinations of products of discrete hypergeometric polynomials; this generalizes relations to discrete orthogonal polynomials for the one-point correlators \langle \tr M^k \rangle of the classical ensembles recently discovered by Cunden et al. [3]. Hence, we turn our attention on the combinatorial interpretation of correlators for the Laguerre and Jacobi ensembles. We prove that the coefficients in the topological expansion of Jacobi correlators are multiparametric single Hurwitz numbers involving combinations of triple monotone Hurwitz numbers. Via a simple limit, this reproduces formulae of [2] on the Laguerre ensemble. This completes the combinatorial interpretation of correlators of unitary ensembles with classical potential. Combining results of Dubrovin et al. [4], and of Norbury [10] connecting integrable systems with enumerative geometry, we obtain ELSV-like formulae linking the multiparametric single Hurwitz numbers of LUE and JUE respectively to cubic Hodge integrals and $\Theta$-GW invariants. In the second part of the thesis we analyse various integrable dynamical systems from a probabilistic point of view. Specifically, we study the spectrum of their random Lax Matrix equipped with the associated Gibbs Measure, in the spirit of [9, 11]. This is the content of the preprint [6], in collaboration with T.~Grava, G.~Gubbiotti and G.~Mazzuca. We explicitly compute the density of states for the exponential Toda lattice and the Volterra lattice showing they are connected to the Laguerre $\beta$-ensemble at high temperatures and the $\beta$-antisymmetric Gaussian ensemble at high temperatures respectively. For generalizations of these system we derive numerically their density of states and compute their ground states. [1] M. Bertola, B. Dubrovin, and D. Yang, Correlation functions of the KdV hierarchy and applications to intersection numbers over Mg,n, Phys. D, 327 (2016), pp. 30–57. [2] F. D. Cunden, A. Dahlqvist, and N. O’Connell, Integer moments of complex Wishart matrices and Hurwitz numbers, Ann. Inst. Henri Poincar ́e D, 8 (2021), pp. 243–268. [3] F. D. Cunden, F. Mezzadri, N. O’Connell, and N. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys, 369 (2019), pp. 1091–1145. [4] B. Dubrovin, S. Q. Liu, D. Yang, and Y. Zhang, Hodge-GUE correspondence and the discrete KdV equation, Comm. Math. Phys, 379 (2020), pp. 461–490. [5] B. Eynard, T. Kimura, and S. Ribault, Random matrices, arXiv preprint arXiv:1510.04430, (2015). [6] M. Gisonni, T. Grava, G. Gubbiotti, and G. Mazzuca, Discrete integrable systems and random Lax matrices, arXiv preprint arXiv:2206.15371, (2022). [7] M. Gisonni, T. Grava, and G. Ruzza, Laguerre ensemble: Correlators, Hurwitz numbers and Hodge integrals, Ann. Henri Poincar ́e, 21 (2020), pp. 3285–3339. [8] M. Gisonni, T. Grava, and G. Ruzza, Jacobi ensemble, Hurwitz numbers and Wilson polynomials, Lett. Math. Phys., 111 (2021), pp. 1–38. [9] T. Grava and G. Mazzuca, Generalized gibbs ensemble of the Ablowitz-Ladik lattice, Circular β-ensemble and double confluent Heun equation, arXiv preprint: 2107.02303, (2021). [10] P. Norbury, Gromov-Witten invariants of P1 coupled to a KdV tau function, Adv. Math., 399 (2022), p. 108227. [11] H. Spohn, Generalized Gibbs ensembles of the classical Toda chain, J. Stat. Phys., 180 (2020), pp. 4–2

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.Comment: 27 page

Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

We consider the Laguerre partition function, and derive explicit generating func-tions for connected correlators with arbitrary integer powers oftraces in terms of products ofHahn polynomials. It was recently proven in [22] that correlators have a topological expansionin terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed fromour formul\ue6. As a second result we identify the Laguerre partition function with only positivecouplings and a special value of the parameter\u3b1= 121/2 with the modified GUE partitionfunction, which has recently been introduced in [28] as a generating function for Hodge inte-grals. This identification provides a direct and new link between monotone Hurwitz numbersand Hodge integrals

A KLT-like construction for multi-Regge amplitudes

Inspired by the calculational steps originally performed by Kawai, Lewellen and Tye, we decompose scattering amplitudes with single-valued coefficients obtained in the multi-Regge-limit of N=4 super-Yang-Mills theory into products of scattering amplitudes with multi-valued coefficients. We consider the simplest non-trivial situation: the six-point remainder function complementing the Bern-Dixon-Smirnov ansatz for multi-loop amplitudes. Utilizing inverse Mellin transformations, all single-valued amplitude components can indeed be decomposed into multi-valued amplitude components. Although the final expression is very similar in structure to the Kawai-Lewellen-Tye construction, moving away from the highly symmetric string scenario comes with several imponderabilities, some of which become more pronounced when considering more than six external legs in the remainder function.Comment: 39 pages, 11 figures, 3 appendice

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172 pages, low quality figures, better quality figures upon request to the authors, comments welcom

The multi-scale nature of the solar wind

The solar wind is a magnetized plasma and as such exhibits collective plasma behavior associated with its characteristic spatial and temporal scales. The characteristic length scales include the size of the heliosphere, the collisional mean free paths of all species, their inertial lengths, their gyration radii, and their Debye lengths. The characteristic timescales include the expansion time, the collision times, and the periods associated with gyration, waves, and oscillations. We review the past and present research into the multi-scale nature of the solar wind based on in-situ spacecraft measurements and plasma theory. We emphasize that couplings of processes across scales are important for the global dynamics and thermodynamics of the solar wind. We describe methods to measure in-situ properties of particles and fields. We then discuss the role of expansion effects, non-equilibrium distribution functions, collisions, waves, turbulence, and kinetic microinstabilities for the multi-scale plasma evolution.Comment: 155 pages, 24 figure

Property Preservation and Quality Measures in Meta-Models.

This thesis consists of three parts. Each part considers different sorts of meta-models. In the first part so-called Sandwich models are considered. In the second part Kriging models are considered. Finally, in the third part, (trigonometric) Polynomials and Rational models are studied.

Improving downstream processing for viral vectors and viral vaccines

Viral vectors are playing an increasingly important role in the vaccine and gene therapy elds. The broad spectrum of potential applications, together with expanding medical markets, drives the e orts to improve the production processes for viral vaccines and viral vectors. Developing countries, in particular, are becoming the main vaccine market. It is thus critical to decrease the cost per dose, which is only achievable by improving the production process. In particular advances in the upstream processing have substantially increased bioreactor yields, shifting the bioprocess bottlenecks towards the downstream processing. The work presented in this thesis aimed to develop new processes for adenoviruses puri cation. The use of state-of-the-art technology combined with innovative continuous processes contributed to build robust and cost-e ective strategies for puri cation of complex biopharmaceuticals.(...