Singularity and regularity of spatial laws in strongly correlated particle systems, from local to global

Strongly correlated particle systems, such as Coulomb gases and zeros of Gaussian polynomials, have been shown in recent years to display a wide range of behavior vis-a-vis their spatial laws. In particular, their local distributions under spatial conditioning are often characterized by intriguing singularity phenomena, in spite of the highly regular nature of their underlying randomness. We will survey the landscape of this interplay between singularity and regularity, as we scale from local to global. At the local scale, we will discuss recent results unveiling residual regularity that complements the singular spatial distributions. Zooming out, we will explore how the influence of local constraints scale up and manifest themselves in the form of global singularities in a large-deviations regime. Time permitting, we will also discuss the effect of background factors, such as the impact of ambient geometry. No prior topical knowledge will be required.

This talk is based in part on the following works (and their references):

1. Gangopadhyay, U., Ghosh, S., Tan, K.A. (2024). Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles. Communications in Pure Appl. Math. 77(8):3427-3519.
2. Ghosh, S., Nishry, A. (2019). Gaussian complex zeros on the hole event: the emergence of a forbidden region. Communications in Pure Appl. Math. 72(1):3-62.
3. Dinh, T.C., Ghosh, S., Wu, H. (2024). Hole event for random holomorphic sections on compact Riemann surfaces. arXiv:2402.11672.