Natural systems consist of many interacting degrees of freedom. The corresponding dynamical behaviour is frequently erratic, i.e. strongly inﬂuenced by minute changes in system’s parameters, like e.g. boundary conditions. In quantum systems this may lead to huge sample-to-sample ﬂuctuations of observable properties while in classical systems to changes in topological characteristic of the dynamical ﬂow. This intrinsic stochasticity and disorder require statistical tools to achieve a quantitative description of the system’s behaviour. In these regards, Random Matrix Theory is the leitmotif of the whole thesis as it provides powerful and versatile techniques for such studies and it connects the investigated topics. The ﬁrst chapter of this thesis contains the relevant analytical results from Random Matrix Theory necessary for the comprehension of the following chapters. Particular emphasis is devoted to averages of characteristic polynomials of diﬀerent random matrix ensembles.
The ﬁrst topic presented in this thesis is the description of quantum scattering in systems with wave chaos. This is an area of active experimental interest and provides one of the best veriﬁcation of Random Matrix Theory. In particular, we will focus on the presence of uniform absorption, with and without the hypothesis of time-reversal invariance. We computed the distributions of the real and imaginary parts for oﬀ-diagonal entries of the Wigner reaction matrix. Such calculations were made possible by previous results for characteristic polynomials of well-known random matrix ensembles and the use of Berezin integrals. We published this work in [1].
The next chapter is devoted to the description of the phase portrait and chaos in classical disordered systems. After a brief overview of recent models and May’s work in 1972, we go beyond the linear approximation supported by the Hartman-Grobman theorem, around equilibrium points. We connect this problem and the notion of topological complexity with the mean number of real roots of random multivariate Kac polynomials. A ﬁne tuning of the free parameters reveals the existence of a ”resilience radius”. Assuming the origin to be stable, the number of ﬁxed points within such radius is exponentially suppressed as the size of the system grows. This represents a measure of the resilience for disordered systems. In ecological terms, we show that the study of resilience of randomly assembled systems has to go through the investigations of higher order interactions among species. This work has culminated into a paper available at [2]. This chapter is ended with three subsections. In the ﬁrst one, we introduce a dynamical mean ﬁeld approach to address and describe the co-existence of ﬁxed points and chaotic motion. However, what presented requires additional work before reaching a publication level. In the second subsection, we investigate the phase portrait for systems whose dynamics is generated by the superposition of random periodic potentials with random amplitudes and wave vectors. This model represents a ﬁrst connection between dynamical systems and the spectrum of generalized Wishart matrices. The coexistence of more sources of disorder has several implications. Qualitative and few analytical results lead to a rather diﬀerent statistical picture compared to the behaviour of the models above. Indeed, the energy landscape does not seem to undergo abrupt topological changes, as for any values of the control parameter, the ﬁxed points remain, on average, exponentially abundant. This project is in preparation for publication [3]. The last subsection contains few remarks on May’s model with delayed response.
Lastly, the project described in chapter 4 remains, at present, to a preliminary stage. The aim of this chapter is the investigation of critical values for a random, constrained and quadratic function. The complexity of this problem is contained in the deﬁnition and tractability of the feasible set. Additionally, diﬃculties reside in the numerical validation of the analytical results obtained by the replica trick.
This thesis does not include the following second-author publications:
G.Gradoni, S.Belga Fedeli, M.Richter, O.Legrand, “Mutual Information Statistics for Wireless MIMO Propagation Channels in Conﬁned Environments”. In ﬁnal review before submission-2021.
Original language | English |
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Award date | 1 Jun 2021 |
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