footnote

In this paper we are concerned with the small-scale structure of random plane waves. Random plane waves are functions $u$ of the following form:

where $\Lambda\subset\mathbb{R}^{n}$ is a set of equally spaced momenta. Exact choices for $\Lambda$ vary somewhat between papers in this field but $\Lambda$ is usually a neighbourhood of the unit sphere $\mathbb{S}^{n-1}$. The randomness is injected through the coefficients $c_{j}$ which are chosen according to a probability distribution. Such functions $u$ were conjectured, by Berry [3], to provide a good model for the behaviour of chaotic modes in billiard systems.

There are a number of interesting probability distributions from which to draw the $c_{j}$. For example we may choose to look at cases where each $c_{j}$ is chosen as an independent random variable such as Gaussian or Rademacher (see for instance [3], [21],[8]). Alternatively in situations where it is preferable to be able to fix the $\ell^{2}$ norm of $c=[c_{1},\dots,c_{|\Lambda|}]^{T}$ the coefficients may be chosen from a uniform probability density on the high dimensional sphere $\mathbb{S}^{|\Lambda|-1}$ (see for example [20],[6],[14],[22] and [11]).

In this paper we treat a model where each $c_{j}$ is a (independent, identically distributed) Gaussian random variable and $\Lambda=\Lambda_{\beta}$ is a set of $h$-separated momenta drawn from $\mathbb{S}^{n-1}\times[1-h^{\beta},1+h^{\beta}]$ for $0\leq\beta\leq 1$. This range of $\beta$ corresponds to the range of random waves for which we have numerical experiments. The website of Alex Barnett [2] records a number of these experiments including videos generated at the AIM workshop Topological complexity of random sets showing the effect of varying $\beta$. It is convenient to choose $\Lambda_{\beta}$ so that the momenta are equally spaced, this is however not strictly necessary. What is necessary is that the spacing between momenta is never less than $h$ (this is an uncertainty principle requirement) and that the number of momenta in a region scales with the volume of that region. To allow normalisation we restrict our attention to the behaviour of $u$ inside the ball of radius one about zero, $B_{1}(0)$. Then the variance for the Gaussian random variables is determined by adopting the convention that

This then means that the variance $\sigma^{2}=\frac{1}{N}$ where $N=|\Lambda|$.

The key questions of quantum chaos (see for example the survey [23]) are concerned with limits of $\langle Pu_{h},u_{h}\rangle$ where $P$ is a semiclassical pseudodifferential operator. A sequence of $u_{h}$ on a compact manifold $M$ with $h\to 0$ is said to equidistribute if

where $S^{\star}M$ is the unit co-tangent bundle equipped with Liouville measure and $p$ is the principal symbol of $P$. This leads to a natural question, are random waves equidistributed?

It is reasonably easy to ascertain that if $p(x,\xi)$ (the symbol of $P$) is independent of $h$ then random waves are indeed equidistributed. A more subtle question pertains to the small-scale structure. If for example $p(x,\xi;h)$ is a symbol that is zero off a small (shrinking in $h$) region of space in $T^{\star}M$ does equidistribution (in an almost sure sense) still hold? If not does a weaker form of equidistribution hold where $\langle Pu,u\rangle$ is only proportional to the phase-space volume of $p(x,\xi;h)?$

Numerical studies of random waves have suggested that at very small scales there are fluctuations that exceed equidistribution. As seen in Figure 1 (reproduced with permission of Alex Barnett) random waves appear to have enhancements along some straight lines. This so named filament structure was intensely discussed at the AIM workshop Topological complexity of random sets (see the report [1]). Some participants taking the opinion that the apparent structure was simply a numerical artefact other believing that it could be quantified. No firm conclusion was reached at the workshop or since. In this paper we take some steps toward understanding what precisely gives rise to this filament structure. As observed in the AIM report [1] the linear structure appears to be constructed of many smaller filaments roughly aligned along a straight lines. The simplest form of failure to equidistribute would then be to have some straight line segments $\gamma$ where $\left|\!\left|{u}\right|\!\right|_{L^{2}}$ was unusually large. Due to VanderKam [19] and Zelditch [20] the maximum fluctuation we could expect would be logarithmic. In this paper we will see that this sort of concentration does not occur. In Theorems 2.1 and 2.2 we show that the X-ray transform of the modulus squared of a random wave is uniformly equidistributed with high probability. That is for $\gamma_{(x,\xi)}$ a unit length line segment, properly contained in $B_{1}(0)$, from $x$ in direction $\xi$ the random variable $\left|\!\left|{u}\right|\!\right|_{L^{2}(\gamma_{(x,\xi)})}=\left(\int_{\gamma_{(x,\xi)}}|u|^{2}dl_{\gamma}\right)^{1/2}$ obeys

and that there is a $\kappa>0$ (details in Theorem 2.2) so that

In terms of what we might expect to see in a numerical simulation the uniformity (in $(x,\xi)$) of the tail bound (0.1) is crucial. If for example we were only able to show that for fixed $(x,\xi)$

we could conclude that if we picked a line segment then considered all potential random waves the chance that we would see a failure to equidistribute on that particular segment is small. For an example of this kind of result see [10] (in the setting where the coefficients are chosen according to a uniform distribution on a high dimensional sphere). Such a statement would not preclude the possibility that most random waves have some $(x,\xi)$ for which $\left|\!\left|{u}\right|\!\right|_{L^{2}(\gamma(x,\xi))}$ is large. Therefore to rule out structure of this form we do indeed need the stronger bound (0.1). In Section 2 we establish Theorem 0.1 which guarantees uniform equidistribution of $\left|\!\left|{u}\right|\!\right|_{L^{2}(\gamma(x,\xi))}$.

It is instructive to compare Theorem 0.1 with general results about restriction of spectral clusters to curves. When $\beta=1$ the requirement that $\xi_{j}\in\Lambda_{\beta}$ ensures that $u$ is a spectral cluster of width one. Therefore we can apply the results of Burq-Gérard-Tvetkov [5] and Hu [13] to obtain, that for any choice of coefficients,

The results of Theorem 0.1 should be interpreted as saying that choices of coefficients $c=[c_{1},\dots,c_{N}]$ that saturate the general bounds are highly unusual.

In the setting of $(M,g)$ a Riemannian manifold with ergodic geodesic flow the quantum ergodic restriction results of [18] and [9] imply that for density one subsequences of exact eigenfunctions $\phi_{j}$,

where $L(\gamma)$ is the length of the curve $\gamma$. The agreement between this and Theorem 0.1 should be seen as evidence of the agreement between quantum ergodic systems and the predictions of the random wave model.

We then turn our attention to the phase-space picture and consider a random variable $G_{\alpha,\beta,\mu}(x,\xi)$ which measures the phase-space concentration of $u$ near the point $(x,\xi)\in\mathbb{R}^{n}\times\mathbb{S}^{n-1}$ given by

Here $P^{\alpha,\mu}_{(x,\xi)}$ is a semiclassical pseudodifferential operator that localises $u$ in phase space near $(x,\xi)$. The parameter $\alpha$ controls the shape of the localisation, the parameter $\mu$ controls the distance from Planck scale. When $\mu=1$ the operator $P^{\alpha,\mu}_{(x,\xi)}$ localises $u$ up to the limits allowed by the uncertainty principle. As $\mu$ increases the localisation becomes more relaxed. Normalisations are chosen so that if $u$ is phase-space equidistributed $|G_{\alpha,\beta,\mu}(x,\xi)|=1+O(\mu^{-1})$. The details of symbol of $P^{\alpha,\mu}$ are described in Section 3. We discover that once $\mu$ is large, that is $\mu\approx h^{-\epsilon})$, we again obtain a uniform equidistribution result similar to that for the X-ray transform of $|u|^{2}$. However when $\mu\leq C$ we not only fail to obtain equidistribution but are able to show that with high probability there are logarithmic enhancements near some points $(x,\xi)$. The intermediate scales $C\leq\mu\leq h^{-\epsilon}$ remain something of a mystery and will likely require very delicate analysis to fully resolve.

Therefore once we are at Planck scale we expect to see some regions of phase space that where $u$ displays concentration that is logarithmically larger than that predicted by equidistribution. This result should be seen as aligning with Burq-Lebeau’s result [6] that

Indeed when $\alpha=0$ the operator $P^{\alpha,\mu}_{(x,\xi)}$ localises $u$ to a small ball of radius $h$ around $x$. Since $u$ is constituted of waves of frequency of size $1/h$ the value of $|u(x)|$ cannot vary much on balls of radius $h$. Taking into account the normalisation we therefore get that

Therefore the results of this paper for $\alpha=0$ should be seen as morally equivalent to the Burq-Lebeau result.

The question then remains, just what are we seeing in numerical simulation that display filament structure? We know from Theorem 0.1 any scaring along a line segment $\gamma(x,\xi)$ cannot be enough to register in the $L^{2}$ norm. However it is quite possible to have large number of logarithmically bright points aligned along a ray without the $L^{2}$ norm along that ray becoming large. To the human eye examining a numerical experiment, such a situation may appear as a scar. This suggests we should look at weaker forms of structure rather than $L^{2}$ scaring. One possibility, suggested by the phase-space failure to equidistribute, is that the bright points tend to align along rays that have a Planck-scale enhancement in the phase-space sense.

Throughout this paper we will use concentration of measure arguments to allow us to “commute the expectation operator with a function $\phi$”. The idea is that if $\phi:[0,\infty)\to[0,\infty)$ is invertible, we will use measure concentration to find (see Section 1) sufficient conditions so that

Here $X$ is a random variable that takes values in $[0,\infty)$. The rational for seeking this kind of estimate is to be able to chose $\phi$ so that $\mathbb{E}[\phi(X)]$ is easy (or at least tractable) to compute. Just by assuming $\phi$ is convex and strictly increasing we can get the upper bound (by Jensen). However, typically, we need the full inequality. In some cases we will in fact be able to find lower order control on $\left|\phi\left(\mathbb{E}[X]\right)-\mathbb{E}\left[\phi(X)\right]\right|$ therefore obtaining control on the “Jensen gap”.