A New Distribution

[This is a relatively technical post.]
The red curve in this figure is the probability distribution of the ground state energy of the microstate spectra of quantum completions of JT gravity. Put differently (the way Wigner might have) if you ask what are all the discrete spectra that are compatible with the leading semi-classical result for quantizing JT gravity (the famous Schwarzian result for the density of states: ), this curve gives the shape of the distribution of ground states. (The blue curve is simply the associated CDF.) I first uncovered this distribution in a paper last year, with further insights and generalizations in a paper earlier this year, along with the distributions for higher energy levels that follow from it. But the exciting new result of my paper from a few weeks ago is that I have now shown that it is a solution of an ordinary differential equation (or a family of them). This allows for some powerful universal things to be said analytically about the properties of the distribution!

This is fully analogous to what happened for the well-known Tracy-Widom distribution for the largest (or smallest) energy of Gaussian random hermitian matrices. While many workers (such as Forrester) had uncovered important aspects of the distribution, and while it was known that it can be expressed as a particular Fredholm determinant, Tracy and Widom broke new ground in 1994 by showing that the distribution was governed by a well known ODE – the Painleve II equation – and in particular can be given in terms of a special solution of it studied earlier by Hastings and McLeod. The result helped forge further connections between properties of random matrix theory and several interesting areas of mathematics and mathematical physics. Furthermore, the TW distribution has been recognized as marking a new kind of universality, showing up in a wide range of key systems in experimental and theoretical physics, and mathematics. As I say in my recent paper, it governs, e.g., the longest increasing subsequence of random permutations of integers, height variances in certain growth processes in the KPZ class, and the onset of instability in certain random dynamical systems. There’s a lovely Quanta magazine article on it and these connections, linked here.

So what’s the big deal about my recent result? Well, it was already extremely exciting to be able to get the distributions of the energy levels of the quantum gravity system by numerically evaluating the Fredholm determinant expressions for them, as I described in my February post. But I’d already been wondering (for over a year now) whether there was a means of deriving the result from a differential equation. This would be a breakthrough since it would allow analytic statements to be made about the behaviour of the distribution’s tails. (In fact, it is such behaviour that allows for experimental comparison, in the case of TW.) Moreover, it would forge connections to other areas, including the characterization of the distribution’s universal features in terms of the neighborhood of a third order phase transition. (In the case of TW and Gaussian matrix models it connects to the Gross-Witten-Wadia transition, and indeed in this gravity case there is a special analogue of that that plays a role.)

There are more reasons this is really exciting, connecting to a new kind of universality (at least for gravity, but I suspect it is more far-reaching). (For some background, see another post I did here.) It has been the case for some years now that, although the JT gravity system (a 2D gravity plus a scalar coupled in a particular way) looks like some very special toy two-dimensional gravity system of interest only to specialists, it is showing up in a wide range of problems in gravity and quantum gravity. The leading low-temperature dynamics of a large class of black holes (and solutions related to black holes) in various dimensions seems to be expressible as a JT gravity system. For example, for a 4D charged black hole, it is well known that the near-horizon geometry when T=0 is a product of 2D AdS and a 2D sphere, where the latter’s size is fixed by the charge of the black hole. The Bekenstein-Hawking entropy, denoted , is simply 1/4 the area of this special sphere.
The non-zero (low) temperature dynamics is controlled by JT gravity, gravity in 2D AdS plus a certain scalar, where the scalar represents the deviation of the size of the 2D sphere away from its extremal value. There are many papers showing that this is not an isolated special case: a number of other systems in various dimensions have a low energy sector that again yield JT, and on solving the physics, the leading dynamics are controlled by a particular effective action of what’s called Schwarzian form, with a particular robust form for the density of states given above. (See my recent essay for a summary of some of this.) The really nice thing about my new results is that the distribution of the allowed ground states encodes this universal Schwarzian behaviour in two distinct lovely ways in the falloff to the left and right. This means that even for the large class of deformations of JT gravity that produce subleading corrections to the Schwarzian form for the spectral density, the tails will still be of the rather robust form I derive in the paper.

Anyway, there’s a lot more to say, but that’s why I wrote the paper, so rather than repeat it here, you can go and read it. It’s only four pages, as I wrote it as a potential PRL submission given the wide range of interesting fields this potentially connects to.

Oh! I have to mention some rather amusing things about all this. It turns out that the key differential equation (or family thereof) that forms the foundation of all of this is again something I’d co-discovered and studied way back in 1992 as part of my thesis work, and discussed in a recent post here. The “string equation” where is a special combination of , the function and its derivatives, and a prime is an -derivative, was derived back in 1991/1992 in work by myself with Dalley and Morris (e.g. here for an early summary). You can interpret it all as telling you about how do define a random matrix ensemble whose lowest energy is . I did lots of further work on it in several papers and applications over the years, using it for a variety of problems. Using various things I noticed in later years (see a paper with students Carlisle and Pennington here here) about how to derive some neat expressions for the wavefunction associated to the lowest energy, you can quickly derive (via some very nice detours that exploit a “Miura’’ map of the above system to a hierarchy of equations generalizing Painleve II, found in 1992 with the same co-authors plus Watterstam) a nice expression for the distribution of the lowest energy of the matrix ensemble. In fact, the result is very general, since the precise form of the object encodes a wide class of models.

Amusingly, if you put in the simplest form , corresponding to the Gaussian hermitian case, you quickly get the Tracy-Widom result and the map to Painleve II, which Tracy and Widom found the following year (so had we thought to ask the question back then we’d have found that we’d already derived all the key equations! Such questions were not on the radar of gravity/string folk back then, so we did not ask that question.) In fact, the equation above was rediscovered by Nadal and Majumdar 20 years later (2011) as the central exhibit in an alternative derivation of Tracy-Widom. Moreover, other workers (Cleays et al, and Akemann and Atkin – see my paper for refs.) built on Nadal and Majumdar and derived equations for higher order analogues of Tracy-Widom that as far as I can tell from what I’ve seen are (in essence) equivalent to this same setup, rediscovering the key 1992 equation above along with structures that amount to simply working out the ground state wave function along the lines I explained in 2005.

(This is not the first time the equation, and the larger family captured by more general has been rediscovered elsewhere in the literature. It turns out to have been identified as a Painleve~XXXIV hierarchy of equations, which I did not know until recently. Once I searched the literature under that name, I see that many of the properties of the system that I discovered (some on my own, some with co-authors) and explored 30 years ago, and 20 years ago, have been completely re-discovered, but sadly now attributed to others… The usual story of my career in this field!)

A core point here is that my new results for gravity are not about any particular higher order analogue of TW (that would not be particularly exciting news on its own in 2022) but an entirely new system pertaining to gravity, that can be considered as a special combination of an infinite number of them (in the same way that JT can be considered as an infinite combination of minimal string models, as I’ve described elsewhere).

Anyway, time is short, so I will stop here.

-cvj