TracyWidom beta=4 seems off by a constant

[alanedelman]

I compared with TracyWidomBeta.jl and also the following random airy operator at beta=4 and those two matched. The code in this package is off by 1.14 or something, if it were the sqrt(2) (1.41 not 1.14) I might have understood why. Perhaps something in the normalization? Happy to understand better.

function stochastic(β,n)
    h=n^-(1/3)
    x=h:h:10
    N=length(x)
    d=(-2/h^2 .- x) + sqrt(4/(h*β))*randn(N) # diagonal
    e=fill(1/h^2,N-1) # subdiagonal
    eigmax(SymTridiagonal(d,e))
end

[dlfivefifty]

I suspect the code is still what @jiahao wrote a decade ago.... so I don't know if anyone will remember why

[alanedelman]

so there is an oddball convention when beta=4 that seems to be out there Bornemann mentions this in his footnote on page 4 of https://arxiv.org/pdf/0904.1581.

That number 1.14 mentioned earlier is in fact 2^(1/6) ≈ 1.12.

This convention is in wikipedia and mathematica.

IMHO, it is a thoroughly ridiculous idea because it doesn't generalize to other betas though there are some arguments in favor of the convention.

[al-Khwarizmi]

There is a generalization to other betas, see Theorem 2 in Forrester, A Random Matrix Decimation Procedure Relating β = 2/(r+1) to β = 2(r+1), Comm. Math. Phys. 285, 653–672 (2009). In fact, this is an example of the general β to 4/β duality, see Desrosiers, Duality in random matrix ensembles for all β, Nuclear Physics B 817 [PM] (2009) 224–251.