When particles interact, say by attracting or repulsing, they tend to form nice distributions as the number of particles become large. Examples include both physical (electrons in a potential well) and biological (flocking birds, bacteria). Naive simulation via differential equations proves insufficient, with computational cost becoming prohibitively expensive in more than one dimensions. Instead, we will introduce techniques based on a measure minimisation reformulation using weighted orthogonal polynomials, where by incorporating the correct singularities of the distributions we can rapidly and accurately compute many such distributions in arbitrary dimensions. This leads to high accuracy confirmation of open conjectures on gap formation (imagine a flock of birds with no density in the middle).
These techniques involve understanding the relationship between orthogonal polynomials and singular integral (Hilbert, Riesz, and log kernel) transforms, which have wide reaching consequences. We further explore connections to orthogonal polynomials and random matrix theory, the numerical solution of partial differential equations using boundary integral reformulation, and fractional differential equations.