A derivation of the Tracy-Widom distribution by the Hankel composition method
A celebrated result in the theory of random matrices is the connection between the extremal eigenvalue of a random matrix sampled from the Gaussian Unitary Ensemble and the Painlevé II equation. In this post I will give a derivation of this that uses the Hankel composition method. This method is given in great generality in the paper of Bothner and was used in our joint work to relate the distribution function of the extremal eigenvalue in the elliptic Ginibre ensemble to an integro-differential equation. By showing how this method works in the case of the GUE it should shed some light on how it works in other cases.
Background
The Gaussian Unitary Ensemble (GUE) is an ensemble of n×n Hermitian random matrices with probability density
ZGUE1e−21tr(H2)
ZGUE is a normalisation constant. We are interested in the distribution of the extremal (rightmost) eigenvalue. A famous result (see Chapter 24 of Mehta’s Random Matrices) shows that the cumulative distribution converges, under an appropriate scaling, to the Fredholm determinant of the Airy kernel. Let λn be the rightmost eigenvalue.
F(t)≡n→∞limP(λn≤2n+2n61t)=det(1−K)L2(t,∞)
where K:L2(t,∞)→L2(t,∞) is the operator with kernel
The motivation for studying this is not simply that the GUE is an easy model to study, but also that this Airy kernel is universal (see Deift’s Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach). That is, suppose we have an ensemble of Hermitian matrices with probability density
ZV1e−ntrV(H)
for some entire function V which grows sufficiently rapidly at ±∞, e.g. a polynomial. For generic V, the eigenvalues will asymptotically (n→∞) concentrate on disjoint intervals [α1,β1],…,[αm,βm]; and the distribution of the extremal eigenvalue at these endpoints α1,β1,…,αm,βm will converge after a suitable rescaling, for “typical” V, to det(1−K)L2(t,∞). There is a similar universality in the bulk where the “universal” kernel is the sine kernel. Gap probabilities in the sine point process were found to be related to the Painlevé V equation by the group of Jimbo, Miwa, Môri and Sato in 1980 (see here for an accessible introduction to this work). The work of Tracy and Widom on the Airy kernel was strongly inspired by the work of this group.
and we have the boundary condition q(t)∼Ai(t) as t→+∞. △
We demonstrate the above by showing that
dt2d2logF(t)=−q(t)2.
We then obtain the above formula by integrating twice. To justify this requires showing that logF(t) and dtdlogF(t) tend to zero at t=+∞. Showing this requires an asymptotic analysis of the Fredholm determinant det(1−K)L2(t,∞) which is beyond the scope of this post.
The first step is to bring the t dependence into the operator. Let
Kt(x,y)=K(x+t,y+t)=∫t∞Ai(x+s)Ai(y+s)ds.
Then F(t)=det(1−K)L2(t,∞)=det(1−Kt)L2(R+).
Notation: We let τt be the shift operator, so that (τtϕ)(x)=ϕ(x+t) and D be the derivative operator, (Dϕ)(x)=ϕ′(x). We shall be somewhat careless and not specify on what spaces these operators act on. Let us also denote the Airy function Ai=A. △
We see that dtdKt(x,y)=−A(x+t)A(y+t). Thus
dtdKt=−τtA⊗τtA.
Remark: I should signpost a point of rigour. We have calculated the derivative with respect to t pointwise on the kernel, but in fact what we’d like is for the limit implicit in the derivative to exist in the trace norm, and a complete proof would show this. △
Remark: Note that trL2(R+)(ψ⊗ϕ)=⟨ψ,ϕ⟩L2(R+)=∫R+ψ(x)ϕ(x)dx (we will always take functions to be real valued). Note also that Kt, and hence (1−Kt)−1, are symmetric operators with respect to this inner product. △
Exercise: The quantity C=p0(t)2−q0(t)2−2p1(t) is conserved. (There are actually infinitely many such conserved quantities but we only need this one.)
Corollary: It seems reasonable that since the Airy function decreases rapidly at +∞ that qn and pn should tend to zero at t→+∞. It therefore follows that C=0. From this it follows that
dt2d2logF(t)=−q0(t)2.
Remark: It is “obvious” that since Kt is “small” for t→+∞
q0(t)≈(τtA)(0)=Ai(t).
This explains the boundary condition. This needs to be rigorously justified but is beyond the scope of this post. △
Closing up the system
Everything up until now has been “universal” – in that we haven’t used any properties of A – we have only used the Hankel composition structure of K. In particular, we haven’t used that A solves the Airy equation, D2A=MA, where M is the operator such that (Mϕ)(x)=xϕ(x). Such a “non-universal” property allows us to close up the system and obtain an ODE for q0. Note that (Mϕ)(0)=0.
From this we get
q2(t)=tq0(t)+([(1−Kt)−1,M]τtA)(0).
As before [(1−Kt)−1,M]=(1−Kt)−1[Kt,M](1−Kt)−1. If we recall our two equivalent formulae for the Airy kernel we see that
[Kt,M]=−τtA⊗DτtA+DτtA⊗τtA.
This gives
q2(t)=tq0(t)+q1(t)p0(t)−q0(t)p1(t).
If we combine this formula with our relation p0(t)2−q0(t)2−2p1(t)=0 we find