Prof. Kanzieper and collaborators' paper was published in PRL.

Professor Eugene Kanzieper, from the Faculty of Sciences at HIT Holon Institute of Technology, published a paper in the prestigious Physical Review Letters journal.
   
Randomness in a classical world is a source of many counter-intuitive statements known as probability paradoxes. They reflect a conflict between our (naive) intuition and probabilistic thinking. The underlying meaning of paradoxes is revealed only by scrutiny. 

The inspection paradox, also known as the bus or the waiting-time paradox, is possibly the most notable and striking example of the counter-intuitive effects of randomness. Documented in the literature over a century ago (1922), it claims that a passenger who arrives at a bus stop at a random time may, on average, wait longer (and sometimes much longer!) than half the average time between consecutive buses. This counter-intuitive statement is rooted in the fact that a passenger is more likely to arrive at a bus stop between two buses that are far apart. This observation is crucial to understanding the inspection paradox. 
 

The “intuitive” answer for the mean waiting time (half the average interval between successive buses) indirectly assumes that traffic is deterministic and not affected by unexpected road disruptions such as traffic jams, road accidents, malfunctioning traffic lights, etc.

( photo: Shutterstock)

 A similar line of reasoning applies to yet another probability paradox – the friendship paradox – discovered in 1991. Summarized in the statement that “on average, your friends have more friends than you do,” the paradox challenges the perceptions of many individuals who tend to believe that they have more friends than their friends. A resolution of the friendship paradox comes from the observation that people with lots of friends are more likely to be among your friends.

The friendship paradox can be used to build an effective vaccination policy by targeting people who are more likely to spread it: https://tinyurl.com/friendship-paradox-video

( photo: Shutterstock)

Mathematically, the bus paradox and the friendship paradox have the same origin: the sampling bias.

In their PRL paper, Prof. Eugene Kanzieper and collaborators bring the inspection paradox to the world of quantum chaos. Defining a notion of local level spacings (which, roughly speaking, is a spectral equivalent of the waiting time; see an explanation in the illustration below), the authors study counter-intuitive statistics of local level spacings using advanced techniques of the Random Matrix Theory. In particular, they show that the mean values of local level spacings are described by distinguished sequences of universal numbers that have been explicitly determined. Measuring these means in an experiment, one may uniquely identify a spectral universality class to which a given single- or many-body quantum chaotic system belongs. The authors provide a proof of concept, by performing high-precision numerical experiments for three paradigmatic systems of quantum chaology: high-lying zeros of the Riemann zeta function on the critical line, spectra of rectangular billiards, and random spectra of the Sachdev-Ye-Kitaev model that has become a popular venue for studies of quantum many-body physics.
 

In the bus paradox (upper panel), a randomly arriving passenger plays a rôle of an inspector measuring a time interval between two neighboring buses – the one that was missed and the one that will be boarded. In random spectra (lower panel), the measured energy levels (spikes on the graph) are equivalent to a sequence of busses; the observation point  φ  mimics the moment of a passenger's arrival at a bus stop, while the distance between consecutive spikes containing the observation point  φ  is the local level spacing.

Abstract of the article in English: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.220401

Statistics of Local Level Spacings in Single- and Many-Body Quantum Chaos 

We introduce a notion of local level spacings and study their statistics within a random-matrix-theory approach. In the limit of infinite-dimensional random matrices, we determine universal sequences of mean local spacings and of their ratios which uniquely identify the global symmetries of a quantum system and its internal—chaotic or regular—dynamics. These findings, which offer a new framework to monitor single- and many-body quantum systems, are corroborated by numerical experiments performed for zeros of the Riemann zeta function, spectra of irrational rectangular billiards, and many-body spectra of the Sachdev-Ye-Kitaev Hamiltonians.
 
This study was undertaken in cooperation with Dr. Peng Tian (research fellow at Université Côte d'Azur, Nice, France) and Dr. Roman Riser (postdoctoral fellow at Texas Tech University, Lubbock, United States). Peng and Roman previously held postdoctoral positions within the School of Mathematical Sciences at H.I.T. The Israel Science Foundation supported the research as part of the project "Deciphering Noise in Spectra of Complex Systems: A Random Matrix Theory Perspective" (ISF Grant No. 428/18).