This is an extended version of my Nobel Lecture, delivered on December 8, 2021. I will
recall the genesis of the concept of multiple equilibria in natural sciences. I will then describe …

In critical systems, the effect of a localized perturbation affects points that are arbitrarily far
from the perturbation location. In this paper, we study the effect of localized perturbations on …

We consider the assignment problem between two sets of N random points on a smooth, two-
dimensional manifold Ω of unit area. It is known that the average cost scales as E Ω (N)∼ …

We discuss the optimal matching solution for both the assignment problem and the matching
problem in one dimension for a large class of convex cost functions. We consider the …

In the previous chapter we presented some random optimization problems on weighted
graphs and some useful tools for their solution. However, we did not discuss the effect of …

Using the replica approach and the cavity method, we study the fluctuations of the optimal
cost in the random-link matching problem. By means of replica arguments, we derive the …

The traveling-salesman problem is one of the most studied combinatorial optimization
problems, because of the simplicity in its statement and the difficulty in its solution. We …

We consider the random Euclidean assignment problem on the line between two sets of N
random points, independently generated with the same probability density function ϱ ϱ. The …

We analytically derive, in the context of the replica formalism, the first finite-size corrections
to the average optimal cost in the random assignment problem for a quite generic …

Given 2n points, n``red''and n``blue'', in a Euclidean space, solving the associated Euclidean
Assignment Problem consists infinding the bijection between red and blue points that …