Universal fluctuations of quasi-optimal paths of the travelling salesman problem

We study the statistical properties of the quasi-optimal solutions to the travelling salesman problem with city positions randomly distributed on a square. To each near-optimal solution we associate points on a circle with the same order and distances. We then analyse the fluctuations of the positions, applying statistical measures developed previously to investigate the behaviour of eigenvalues of (unitary) random matrices. We establish that, in the limit of a large number of cities, these measures display a universal behaviour, intermediate between that of a sequence of uncorrelated random points and a sequence of eigenvalues of unitary symmetric random matrices.