Abstract:
This thesis is based on the article [16], which studies the integral? ? ?a? ?b? s ρ(x1,...,xN) max|xi −xj| min|xi −xj| |xi −xj| ij dx1 ...dxN
KN i<j i<j i<j
where K is an arbitrary p-field, ρ is a well-behaved function that depends only
on the norm of (x1,...,xN) ∈ KN, and a,b,sij are certain complex numbers.
A mixture of analysis and combinatorics is used to find two explicit formulas for the integral (one for b ̸= 0 and one for b = 0) and an explicit description of all
sij ∈ C for which it converges absolutely (for fixed ρ, a, and b). The integral’s role as the canonical partition function for a log-Coulomb gas (in K) is highlighted throughout, leading to a p-field analogue of Mehta’s Integral Formula and formulas for the joint moments of the gas’ diameter and minimum particle spacing. The notion of log-Coulomb gas in P1(K) is also addressed and related to that in K in
a concrete way: The grand canonical partition function for a log-Coulomb gas in P1(K) is the (q + 1)th power of the grand canonical partition function for a log- Coulomb gas in the open unit ball of K.