Málstofa: Ali Khezeli

Fyrirlesari: Ali Khezeli (INRIA, Paris)
Titill: On the Existence of Balancing Allocations and Factor Point Processes
20. október, kl. 11:40. Stofa N-130 í Öskju.

Ágrip: A balancing allocation between measures \(\Phi\) and \(\Psi\) on \(\mathbb R^d\) is a function \(f\) on \(\mathbb R^d\) that transports \(\Phi\) to \(\Psi\). In the case of stationary random measures or point processes, which always have infinite total mass, the study of balancing allocations and balancing transport kernels has been of great interest in the last two decades. This was started by the novel work [Hoffman, Holroyd and Peres, 2005], which provided a generalization of the stable marriage algorithm and constructed a balancing allocation between the Lebesgue measure and the Poisson point process. Balancing transports exist in general, however, for the existence of balancing allocations, no simple necessary and sufficient condition is known yet (under a measurability and translation-invariance condition). Recently, [Last, Thorisson, 2023] provided the sufficient condition that \(\Phi\) is diffuse (i.e., has no atoms) and there exists an auxiliary point process as a factor of \((\Phi,\Psi)\). It was conjectured in [Haji-Mirsadeghi, Khezeli, 2016] that there always exists a factor point process provided that \((\Phi,\Psi)\) has no nontrivial symmetry a.s. In this talk, we prove this conjecture by using the results of the theories of group actions and Borel equivalence relations.

Based on joint work with Samuel Mellick.