Discussion concerning certain details of the page

"The Gaussian Unitary Ensemble Hypothesis"


Mike Hardy wrote:

http://www.maths.ex.ac.uk/~mwatkins/zeta/bump-gue.htm

At the page whose URL is above, you wrote:

> gives the probability density of finding a second
> eigenvalue near t+x. This function is:

and there follows a function that is clearly neither a probability density function (since it does not approach 0 at infinity) nor a cumulative probability distribution function (since it is not monotone). You must have intended something else. Can you clarify? Thanks.

 

Dan Bump (who effectively authored that page) clarified:

It is not a probability distribution. If you wanted the probability density that the NEXT eigenvalue would
be at x+t, that would be a probability distribution, and would go to zero (as you say) as t--> infinity. Just
SOME eigenvalue. The eigenvalues are uncorrelated at large distance but over a short distance tend to repel.

The pair correlation function 1-sin^2(pi x)/(pi x)^2 is actually the limiting distribution as N-->infinity, not exactly right for any fixed N. Also, the eigenvalue density for the GUE is not really constant, so you have to compensate for this by scaling, or restrict your attention range that is of moderate length depending on N.

With these caveats, and for the purposes of discussion we can imagine that there are an infinite number of eigenvalues, and on a SMALL interval I, given no other information but the length of
I, the probability of an eigenvalue in the interval is proportional to the length of the interval.

If we know there is an eigenvalue at x, then this statement is modified depending on the location of I with respect to x. If the interval is far from x, the probability that it contains an eigenvalue is still proportional to the length of the interval. But if it is close to x, the probability is less. In short, there is a tendency of the eigenvalues to repel, and this is captured in the pair correlation function.

 

Mike Hardy replied:

> The pair correlation function 1-sin^2(pi x)/(pi x)^2 is actually
> the limiting distribution as N-->infinity, not exactly right for
> any fixed N. Also, the eigenvalue density for the GUE is not

It had seemed as if you were saying it is a _probability_ density_function_. In fact, Matthew Watkins' page, based loosely on yours, explicitly said it's a probability density function. But here you call it a "pair correlation function", which sounds like something else. You say it's a "limiting distribution", but in what sense is it a "distribution" at all? It is not a probability density function, nor a cumulative
probability distribution function.

> If we know there is an eigenvalue at x, then this statement
> is modified depending on the location of I with respect to x.
> If the interval is far from x, the probability that it contains
> an eigenvalue is still proportional to the length of the
> interval. But if it is close to x, the probability is less.
> In short, there is a tendency of the eigenvalues to repel,
> and this is captured in the pair correlation function.

OK, now you make it sound a but like a _conditional_ probability_density. But it's still not a density. Could it be a factor by which one marginal density is multiplied to get the conditional density?

 

Bump further clarified:

> It had seemed as if you were saying it is a _probability_
> _density_function_. In fact, Matthew Watkins' page, based
> loosely on yours, explicitly said it's a probability density
> function. But here you call it a "pair correlation function",
> which sounds like something else. You say it's a "limiting
> distribution", but in what sense is it a "distribution" at
> all? It is not a probability density function, nor a cumulative
> probability distribution function.

It is exactly the pair correlation function for the GUE. It has a probabilistic interpretation, namely if there is an eigenvalue at x, then the probability of an eigenvalue in the interval (x+t,x+t+epsilon) is about epsilon(1-sin^2(pi t)/(pi t)^2) when epsilon is very small compared with t.

If this doesn't make sense, please look it up somewhere. These things are explained in Mehta's book, or various expository papers such as this paper of Conrey: http://front.math.ucdavis.edu/math.NT/0005300 which discusses the pair correlation function for the zeros of the Riemann zeta function, but the principle is the same. Or Dyson's papers, etc.


 


D. Bump's page is at http://match.Stanford.EDU/rh/gue.html
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