Buongiorno a tutti,
Vorremmo segnalarvi che questo venerdì (24 Novembre) alle ore 14:30 in aula
1BC50 (Torre Archimede, Università di Padova) ci sarà un seminario per il
ciclo di seminari in Probabilità e Finanza di:
*Dario Trevisan* (Università di Pisa)
*Title*: On the Concave One-Dimensional Random Assignment Problem:
Kantorovich Meets Young
*Date*: November 24, 2023 at 14:30, room 1BC50, Torre Archimede, Padova
*Abstract*: We consider the assignment (or bipartite matching) problem
between n source points and n target points on the real line, where the
assignment cost is a concave power of the distance, |x-y|^p, for 0<p<1. It
is known that, differently from the convex case (p>1) where the solution is
rigid, i.e. it does not depend on p, in the concave case it may varies with
p and exhibit interesting long-range connections, making it more
appropriate to model realistic situations, e.g. in economics and biology.
In the random version of the problem, the points are samples of i.i.d.
random variables, and one is interested in typical properties as the sample
size n grows. Barthe and Bordenave in 2013 proved asymptotic upper and
lower bounds in the range 0<p<1/2, which they conjectured to be sharp.
Bobkov and Ledoux, in 2020, using optimal transport and Fourier-analytic
tools, determined explicit upper bounds for the average assignment cost in
the full range 0<p<1, naturally yielding to the conjecture that a "phase
transition" occurs at p=1/2. We settle affirmatively both conjectures. The
novel mathematical tool that we develop, and may be of independent
interest, is a formulation of Kantorovich problem based on Young
integration theory, where the difference between two measures is replaced
by the weak derivative of a function with finite q-variation.
Joint work with M. Goldman (arXiv:2305.09234).
Vi aspettiamo numerosi!
Alberto Chiarini e Alekos Cecchin