Matteo D’Achille (Université Paris-Est Créteil)

One dimensional Euclidean Random Assignment Problems: anomalous scaling and critical hyperbolae

A Euclidean Random Assignment Problem (ERAP in short) is defined as follows. Take two $n$-samples $\mathcal{B} =(b_i)_{i=1}^n$ (blue points) and $\mathcal{R}=(r_j)_{j=1}^n$ (red points) of i.i.d. random variables valued on a metric space $(\Omega,D)$ of dimension $d$ according to a prob. measure $\vu$ (called disorder). For a permutation $\pi : \mathcal{B} \rightarrow \mathcal{R}$, let $\mathcal{H}(\pi)=\sum_{i=1}^n D(b_i,r_{\pi(i)})^p$ be a Hamiltonian depending on the energy-distance exponent $p\in \mathbb{R}$. Consider the ground state energy defined as the random variable $H_{\rm opt} = \min_\pi \mathcal{H}(\pi)$ and let $E(n)$ be the expectation of $H_{\rm opt}$ w.r.t. $\nu$.

Depending on the choice of $(\Omega,D)$, on the disorder $\nu$ and $p$, what is the asymptotic behaviour of $E(n)$ for $n$ large?

In this seminar I will
1) review the state of the art on one-dimensional ERAPs;
2) recall the notions of bulk and anomalous scalings of $E(n)$ in presence of a non-uniform disorder;
3) show how, quite generically, anomalous and bulk scalings can be understood in terms of critical hyperbolae coupling a parameter describing the local behavior of the density of $\nu$ in the region of low density of blue and red points to the exponent p. If time allows, I will also provide some research perspective.

The seminar will be mostly based on a paper (currently in preparation) in collaboration with Andrea Sportiello (CNRS and Université Sorbonne Paris Nord) and issued from my PhD thesis (see https://tel.archives-ouvertes.fr/tel-03098672v1, § 2.6).