README.md

bessel_gumbel

This small project (for now the Mathematica notebook tracy_widom_bessel_gumbel.nb) numerically checks that the Fredholm determinant of a modified Bessel(0) kernel in exponential coordinates (arising as a gap probability in certain last passage percolation models) coincides with the standard Gumbel distribution (see here https://en.wikipedia.org/wiki/Gumbel_distribution for a definition).

The Fredholm determinant (gap probability) in question is computed using the method of Bornemann (On the Numerical Evaluation of Fredholm Determinants, arXiv:0804.2543 [math.NA], available at https://arxiv.org/abs/0804.2543). The modified Bessel kernel is a version of the random matrix theory hard-edge Bessel kernel of Tracy--Widom (Level-Spacing Distributions and the Bessel Kernel, arXiv:hep-th/9304063, available at https://arxiv.org/abs/hep-th/9304063).

This observation appeared first in a paper of Johansson (On some special directed last-passage percolation models, arXiv:math/0703492, available at https://arxiv.org/abs/math/0703492). It appears as a remark without proof, see the second equation below (1.8). Note: I believe that said equation should read (in LaTeX notation) $U_{-1/2} = ...$ as it is the right-hand side of equation (1.8) for which the Bessel parameter is equal to 0 (see the code). This was further elaborated in these two papers (one being an extended FPSAC 2021 abstract of the other, longer version) by Alessandra Occelli and myself:

• Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles (arXiv:2011.07890 [math.CO], available at https://arxiv.org/abs/2011.07890) and
• Discrete and continuous Muttalib--Borodin processes I: the hard edge (arXiv:2010.15529 [math.PR], available at https://arxiv.org/abs/2010.15529)

as can be seen in e.g. Remark 4 of the first reference from the list.