## Periodic Reporting for period 1 - QUERWG (Quantum Unique Ergodicity and Random Waves on Graphs)

Reporting period: 2016-09-01 to 2018-08-31

"Quantum Chaos is a central field in mathematics and physics that emerged in the 1970s. Its original aim is to understand the relationship between quantum mechanics and chaos theory. Chaos theory studies dynamical systems that are highly sensitive to initial conditions. A good way to think about it is the butterfly effect: the idea that, for example, ""a butterfly flapping its wings in China can cause a hurricane in Texas"". On the other hand quantum mechanics describes physical systems on microscopic scales. When the energy of the system becomes very high, in a limit called ""semi-classical"", quantum systems behave almost like classical systems. At this boundary between quantum and classical descriptions one can try to make a connection between the two theories and look for signatures of chaos in quantum systems.

Let us think of a particle evolving in a bounded region of space such as a box or any other type of domain, bouncing against the edges. If the system is chaotic we can imagine that the trajectories followed by such a particle will be unpredictable and pass by most locations inside the domain in a random way. The state of the particle in quantum mechanics is not described by a dot following a precise trajectory, but rather by a wave whose amplitude gives an indication on where it is most likely to find the particle. This wave can be concentrated or delocalised. In chaotic systems, it has been observed that it is mostly delocalised at high energies/frequencies (meaning that the particle can be found uniformly anywhere) but can still concentrate for a few high energy levels. Moreover, on smaller scales it seems that the quantum states at these high energies have some universal description: they resemble a random field following a Gaussian probability distribution. These observations are still not well understood theoretically and the goal of this research is to make progress in the mathematical understanding of these phenomena. As often in mathematics, the abstract model we study goes beyond quantum mechanics and describes waves in a more general way.

In this project we considered an original point of view. Instead of looking at systems with higher and higher frequencies, we can keep the frequency fixed and zoom out, looking at larger and larger scale systems. This zooming out process leads to a type of semi-classical limit, and to observations that are comparable to the ones made in high frequencies. This alternative point of view originally appeared in the study of discrete quantum systems, or in other words systems that can be described by a network. The main achievement of this project was to connect the discrete and continuous descriptions by providing a unifying framework for large scale structures. This new framework generalises the usual high-frequency limit, and requires the development of new mathematical tools, which was started during the project.

As this research is concerned with fundamental science, one of its main impact is on knowledge and the understanding of basic principles in physics and mathematics. It increases our understanding of the solutions of fundamental equations in physics, describing waves and quantum mechanics, and of large networks, giving us new ways to make sense of these objects of major importance in today's world."

Let us think of a particle evolving in a bounded region of space such as a box or any other type of domain, bouncing against the edges. If the system is chaotic we can imagine that the trajectories followed by such a particle will be unpredictable and pass by most locations inside the domain in a random way. The state of the particle in quantum mechanics is not described by a dot following a precise trajectory, but rather by a wave whose amplitude gives an indication on where it is most likely to find the particle. This wave can be concentrated or delocalised. In chaotic systems, it has been observed that it is mostly delocalised at high energies/frequencies (meaning that the particle can be found uniformly anywhere) but can still concentrate for a few high energy levels. Moreover, on smaller scales it seems that the quantum states at these high energies have some universal description: they resemble a random field following a Gaussian probability distribution. These observations are still not well understood theoretically and the goal of this research is to make progress in the mathematical understanding of these phenomena. As often in mathematics, the abstract model we study goes beyond quantum mechanics and describes waves in a more general way.

In this project we considered an original point of view. Instead of looking at systems with higher and higher frequencies, we can keep the frequency fixed and zoom out, looking at larger and larger scale systems. This zooming out process leads to a type of semi-classical limit, and to observations that are comparable to the ones made in high frequencies. This alternative point of view originally appeared in the study of discrete quantum systems, or in other words systems that can be described by a network. The main achievement of this project was to connect the discrete and continuous descriptions by providing a unifying framework for large scale structures. This new framework generalises the usual high-frequency limit, and requires the development of new mathematical tools, which was started during the project.

As this research is concerned with fundamental science, one of its main impact is on knowledge and the understanding of basic principles in physics and mathematics. It increases our understanding of the solutions of fundamental equations in physics, describing waves and quantum mechanics, and of large networks, giving us new ways to make sense of these objects of major importance in today's world."

The phenomenon that in chaotic systems most high-frequency waves delocalise is called the Quantum Ergodicity. The random behaviour of the waves is called the Random Wave Hypothesis. During the project, we proved a quantum ergodicity theorem in large scale limits and made substantial progress in our understanding of the random wave hypothesis. The main achievements were the proof of several delocalisation results for waves in large scale limits on networks and in continuous spaces, and the unification of the continuous and discrete theories via the introduction of a new notion of convergence called the Benjamini-Schramm limit. This limit also puts in the same framework the high-frequency and the large scale limit, and allowed us to formulate rigorously the random wave hypothesis and prove some weak forms of it.

The unification of the discrete and continuous theory through a new formalism (called Benjamini-Schramm convergence) brings new perspectives in the field and paves the way for a rigorous mathematical proof of the observation that waves in chaotic environments behave in a random way, the random wave hypothesis.