0. The effort to
interpret his replica symmetry breaking (RSB) led him and his collaborators M´ezard and Virasoro to predict the existence of a new surprising phase, with broken ergodicity, hierarchically
organised pure states and non-trivial fluctuations of intensive thermodynamic quantities.
The mathematisation of these pioneering results required 30 years of efforts and mathematicians are now discovering vast extensions of these concepts. The impact in physics and other
areas of science was immediate.
The newly gained confidence in the replica method allowed in the 80's to solve Hopfield
model of a neural network and a myriad of applications in neural networks and machine
learning appeared.
In the same years, the RSB based random first order transition theory (RFOT) for structural
glasses was proposed. From the 90's the deeply innovative extensions of the cavity and replica
method to models on Bethe lattices, led to spectacular advances in computer science. Parisi
with M´ezard and Zecchina could exactly find the satisfiability threshold in the celebrated random K-Sat model, and, most interestingly, exploited this exact solution to build a new family
of algorithms that improved the state of the art by several orders of magnitude.
More recently the RSB approach allowed the exact description of glassy phases of particles
in the limit of high dimension. This is the last step in Parisi's long quest for a first principle
theory of structural glasses. This solution provides the microscopic foundation of RFOT theory.
At the same time it proposes a picture that goes much beyond, predicting the existence of a new
fundamental glass-to-glass transition at low temperature where the glass becomes marginally
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J. Phys. A: Math. Theor. 53 (2020) 500301 Preface
stable and ungapped excitations appear. Numerical evidence in favour of the presence of this
transition in finite dimension has been found by Parisi and collaborators. The most spectacular
tested consequence of the theory is a universal description of jammed states of hard spheres,
which accounts for the behaviour found in numerical simulations (also by the group of Parisi) in
spatial dimension D spanning from 2 to 8. While the infinite dimensional solution will remain
a cornerstone in the theory of the glasses the derivation of all its implication is only at the
beginning.
The present special issue celebrates Giorgio Parisi's 70th birthday, and tries to give an
overview of the current state of research in the fields of statistical mechanics and interdisciplinary applications which have been marked by Giorgio Parisi's seminal contributions. The
manuscripts accepted for publication in this special issue do indeed cover a broad range of
subjects, but they share the same theoretical approach based on the techniques pioneered by Giorgio Parisi]]>