Aspects of spatially correlated random fields: Extreme-value statistics and clustering properties

Rare events of large-scale spatially correlated exponential random fields are studied. The influence of spatial correlations on clustering and nonsphericity is investigated. The size of the performed simulations permits to study beyond-7.5-sigma events (one in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mn>10</a:mn> <a:mn>13</a:mn> </a:msup> </a:math> ). As an application, this allows to resolve individual Hubble patches which fulfil the condition for primordial black hole formation. It is argued that their mass spectrum is drastically altered due to cocollapse of clustered overdensities as well as the mutual threshold-lowering through the latter. Furthermore, the corresponding nonsphericities may imply possibly large changes in the initial black hole spin distribution.