Rigidity of Random Point Configurations
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Johnson (1R3 - 1st floor)
A point configuration is a discrete set of points, living in a space, such as R, C, or R^d. There are several ways to generate point configurations, (1) zeros of polynomial or holomorphic functions; (2) eigenvalues of matrices; (3) configurations minimizing certain energies. (Among others!)
These configurations become random as soon as randomness is introduced (obviously). For instance, one may consider random coefficients in (1) or (2), or introduce a strictly positive temperature T in (3) (leading to Boltzmann or Gibbs measures in statistical physics).
One may also choose several points independently and uniformly in a given compact set. In that case, there is no interaction between points: two of them can be arbitrarily close. The three models described above behave very differently and distribute points much more homogeneously in space. They naturally exhibit properties of repulsion and rigidity.
My PhD thesis deals with the rigidity of models of type (3), involving energies with long-range pairwise interactions.