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Quaternions and spherical trigonometry

19 December, 2024 in expository, math.MG | Tags: , , | by | 29 comments

Hamilton’s quaternion number system {\mathbb{H}} is a non-commutative extension of the complex numbers, consisting of numbers of the form {t + xi + yj + zk} where {t,x,y,z} are real numbers, and {i,j,k} are anti-commuting square roots of {-1} with {ij=k}, {jk=i}, {ki=j}. While they are non-commutative, they do keep many other properties of the complex numbers:

  • Being non-commutative, the quaternions do not form a field. However, they are still a skew field (or division ring): multiplication is associative, and every non-zero quaternion has a unique multiplicative inverse.
  • Like the complex numbers, the quaternions have a conjugation

    \displaystyle \overline{t+xi+yj+zk} := t-xi-yj-zk,

    although this is now an antihomomorphism rather than a homomorphism: {\overline{qr} = \overline{r}\ \overline{q}}. One can then split up a quaternion {t + xi + yj + zk} into its real part {t} and imaginary part {xi+yj+zk} by the familiar formulae

    \displaystyle \mathrm{Re} q := \frac{q + \overline{q}}{2}; \quad \mathrm{Im} q := \frac{q - \overline{q}}{2}

    (though we now leave the imaginary part purely imaginary, as opposed to dividing by {i} in the complex case).
  • The inner product

    \displaystyle \langle q, r \rangle := \mathrm{Re} q \overline{r}

    is symmetric and positive definite (with {1,i,j,k} forming an orthonormal basis). Also, for any {q}, {q \overline{q}} is real, hence equal to {\langle q, q \rangle}. Thus we have a norm

    \displaystyle |q| = \sqrt{q\overline{q}} = \sqrt{\langle q,q \rangle} = \sqrt{t^2 + x^2 + y^2 + z^2}.

    Since the real numbers commute with all quaternions, we have the multiplicative property {|qr| = |q| |r|}. In particular, the unit quaternions {U(1,\mathbb{H}) := \{ q \in \mathbb{H}: |q|=1\}} (also known as {SU(2)}, {Sp(1)}, or {Spin(3)}) form a compact group.
  • We have the cyclic trace property

    \displaystyle \mathrm{Re}(qr) = \mathrm{Re}(rq)

    which allows one to take adjoints of left and right multiplication:

    \displaystyle \langle qr, s \rangle = \langle q, s\overline{r}\rangle; \quad \langle rq, s \rangle = \langle q, \overline{r}s \rangle

  • As {i,j,k} are square roots of {-1}, we have the usual Euler formulae

    \displaystyle e^{i\theta} = \cos \theta + i \sin \theta, e^{j\theta} = \cos \theta + j \sin \theta, e^{k\theta} = \cos \theta + k \sin \theta

    for real {\theta}, together with other familiar formulae such as {\overline{e^{i\theta}} = e^{-i\theta}}, {e^{i(\alpha+\beta)} = e^{i\alpha} e^{i\beta}}, {|e^{i\theta}| = 1}, etc.
We will use these sorts of algebraic manipulations in the sequel without further comment.

The unit quaternions {U(1,\mathbb{H}) = \{ q \in \mathbb{H}: |q|=1\}} act on the imaginary quaternions {\{ xi + yj + zk: x,y,z \in {\bf R}\} \equiv {\bf R}^3} by conjugation:

\displaystyle v \mapsto q v \overline{q}.

This action is by orientation-preserving isometries, hence by rotations. It is not quite faithful, since conjugation by the unit quaternion {-1} is the identity, but one can show that this is the only loss of faithfulness, reflecting the well known fact that {U(1,\mathbb{H}) \equiv SU(2)} is a double cover of {SO(3)}.

For instance, for any real {\theta}, conjugation by {e^{i\theta/2} = \cos(\theta/2) + i \sin(\theta/2)} is a rotation by {\theta} around {i}:

\displaystyle e^{i\theta/2} i e^{-i\theta/2} = i \ \ \ \ \ (1)

\displaystyle e^{i\theta/2} j e^{-i\theta/2} = \cos(\theta) j + \sin(\theta) k \ \ \ \ \ (2)

\displaystyle e^{i\theta/2} k e^{-i\theta/2} = \cos(\theta) k - \sin(\theta) j. \ \ \ \ \ (3)

 Similarly for cyclic permutations of {i,j,k}. The doubling of the angle here can be explained from the Lie algebra fact that {[i,j]=ij-ji} is {2k} rather than {k}; it also closely related to the aforementioned double cover. We also of course have {U(1,\mathbb{H})\equiv Spin(3)} acting on {\mathbb{H}} by left multiplication; this is known as the spinor representation, but will not be utilized much in this post. (Giving {\mathbb{H}} the right action of {{\bf C}} makes it a copy of {{\bf C}^2}, and the spinor representation then also becomes the standard representation of {SU(2)} on {{\bf C}^2}.)

Given how quaternions relate to three-dimensional rotations, it is not surprising that one can also be used to recover the basic laws of spherical trigonometry – the study of spherical triangles on the unit sphere. This is fairly well known, but it took a little effort for me to locate the required arguments, so I am recording the calculations here.

The first observation is that every unit quaternion {q} induces a unit tangent vector {qj\overline{q}} on the unit sphere {S^2 \subset {\bf R}^3}, located at {qi\overline{q} \in S^2}; the third unit vector {qk\overline{q}} is then another tangent vector orthogonal to the first two (and oriented to the left of the original tangent vector), and can be viewed as the cross product of {qi\overline{q} \in S^2} and {qj\overline{q} \in S^2}. Right multplication of this quaternion then corresponds to various natural operations on this unit tangent vector:

  • Right multiplying {q} by {e^{i\theta/2}} does not affect the location {qi\overline{q}} of the tangent vector, but rotates the tangent vector {qj\overline{q}} anticlockwise by {\theta} in the direction of the orthogonal tangent vector {qk\overline{q}}, as it replaces {qj\overline{q}} by {\cos(\theta) qj\overline{q} + \sin(\theta) qk\overline{q}}.
  • Right multiplying {q} by {e^{k\theta/2}} advances the tangent vector by geodesic flow by angle {\theta}, as it replaces {qi\overline{q}} by {\cos(\theta) qi\overline{q} + \sin(\theta) qj\overline{q}}, and replaces {qj\overline{q}} by {\cos(\theta) qj\overline{q} - \sin(\theta) qi\overline{q}}.

Now suppose one has a spherical triangle with vertices {A,B,C}, with the spherical arcs {AB, BC, CA} subtending angles {c, a, b} respectively, and the vertices {A,B,C} subtending angles {\alpha,\beta,\gamma} respectively; suppose also that {ABC} is oriented in an anti-clockwise direction for sake of discussion. Observe that if one starts at {A} with a tangent vector oriented towards {B}, advances that vector by {c}, and then rotates by {\pi - \beta}, the tangent vector now at {B} and pointing towards {C}. If one advances by {a} and rotates by {\pi - \gamma}, one is now at {C} pointing towards {A}; and if one then advances by {b} and rotates by {\pi - \alpha}, one is back at {A} pointing towards {B}. This gives the fundamental relation

\displaystyle e^{kc/2} e^{i(\pi-\beta)/2} e^{ka/2} e^{i(\pi-\gamma)/2} e^{kb/2} e^{i(\pi-\alpha)/2} = 1 \ \ \ \ \ (4)

relating the three sides and three equations of this triangle. (A priori, due to the lack of faithfulness of the {U(1,\mathbb{H})} action, the right-hand side could conceivably have been {-1} rather than {1}; but for extremely small triangles the right-hand side is clearly {1}, and so by continuity it must be {1} for all triangles.) Indeed, a moments thought will reveal that the condition (4) is necessary and sufficient for the data {a,b,c,\alpha,\beta,\gamma} to be associated with a spherical triangle. Thus one can view (4) as a “master equation” for spherical trigonometry: in principle, it can be used to derive all the other laws of this subject.

Remark 1 The law (4) has an evident symmetry {(a,b,c,\alpha,\beta,\gamma) \mapsto (\pi-\alpha,\pi-\beta,\pi-\gamma,\pi-a,\pi-b,\pi-c)}, which corresponds to the operation of replacing a spherical triangle with its dual triangle. Also, there is nothing particularly special about the choice of imaginaries {i,k} in (4); one can conjugate (4) by various quaternions and replace {i,k} here by any other orthogonal pair of unit quaternions.

Remark 2 If we work in the small scale regime, replacing {a,b,c} by {\varepsilon a, \varepsilon b, \varepsilon c} for some small {\varepsilon>0}, then we expect spherical triangles to behave like Euclidean triangles. Indeed, (4) to zeroth order becomes

\displaystyle e^{i(\pi-\beta)/2} e^{i(\pi-\gamma)/2} e^{i(\pi-\alpha)/2} = 1

which reflects the classical fact that the sum of angles of a Euclidean triangle is equal to {\pi}. To first order, one obtains

\displaystyle c + a e^{i(\pi-\gamma)/2} e^{i(\pi-\alpha)/2} + b e^{i(\pi-\alpha)/2} = 0

which reflects the evident fact that the vector sum of the sides of a Euclidean triangle sum to zero. (Geometrically, this correspondence reflects the fact that the action of the (projective) quaternion group on the unit sphere converges to the action of the special Euclidean group {SE(2)} on the plane, in a suitable asymptotic limit.)

The identity (4) is an identity of two unit quaternions; as the unit quaternion group {U(1,\mathbb{H})} is three-dimensional, this thus imposes three independent constraints on the six real parameters {a,b,c,\alpha,\beta,\gamma} of the spherical triangle. One can manipulate this constraint in various ways to obtain various trigonometric identities involving some subsets of these six parameters. For instance, one can rearrange (4) to get

\displaystyle e^{i(\pi-\beta)/2} e^{ka/2} e^{i(\pi-\gamma)/2} = e^{-kc/2} e^{-i(\pi-\alpha)/2} e^{-kb/2}. \ \ \ \ \ (5)

Conjugating by {i} to reverse the sign of {k}, we also have

\displaystyle e^{i(\pi-\beta)/2} e^{-ka/2} e^{i(\pi-\gamma)/2} = e^{kc/2} e^{-i(\pi-\alpha)/2} e^{kb/2}.

Taking the inner product of both sides of these identities, we conclude that

\displaystyle \langle e^{i(\pi-\beta)/2} e^{ka/2} e^{i(\pi-\gamma)/2}, e^{i(\pi-\beta)/2} e^{-ka/2} e^{i(\pi-\gamma)/2} \rangle

is equal to

\displaystyle \langle e^{-kc/2} e^{-i(\pi-\alpha)/2} e^{-kb/2}, e^{kc/2} e^{-i(\pi-\alpha)/2} e^{kb/2} \rangle.

Using the various properties of inner product, the former expression simplifies to {\mathrm{Re} e^{ka} = \cos a}, while the latter simplifies to

\displaystyle \mathrm{Re} \langle e^{-i(\pi-\alpha)/2} e^{-kb} e^{i(\pi-\alpha)/2}, e^{kc} \rangle.

We can write {e^{kc} = \cos c + (\sin c) k} and

\displaystyle e^{-i(\pi-\alpha)/2} e^{-kb} e^{i(\pi-\alpha)/2} = \cos b - (\sin b) (\cos(\pi-\alpha) k + \sin(\pi-\alpha) j)

so on substituting and simplifying we obtain

\displaystyle \cos b \cos c + \sin b \sin c \cos \alpha = \cos a

which is the spherical cosine rule. Note in the infinitesimal limit (replacing {a,b,c} by {\varepsilon a, \varepsilon b, \varepsilon c}) this rule becomes the familiar Euclidean cosine rule

\displaystyle a^2 = b^2 + c^2 - 2bc \cos \alpha.

In a similar fashion, from (5) we see that the quantity

\displaystyle \langle e^{i(\pi-\beta)/2} e^{ka/2} e^{i(\pi-\gamma)/2} i e^{-i(\pi-\gamma)/2} e^{-ka/2} e^{-i(\pi-\beta)/2}, k \rangle

is equal to

\displaystyle \langle e^{-kc/2} e^{-i(\pi-\alpha)/2} e^{-kb/2} i e^{kb/2} e^{i(\pi-\alpha)/2} e^{kc/2}, k \rangle.

The first expression simplifies by (1) and properties of the inner product to

\displaystyle \langle e^{ka/2} i e^{-ka/2}, e^{-i(\pi-\beta)/2} k e^{i(\pi-\beta)/2} \rangle,

which by (2), (3) simplifies further to {\sin a \sin \beta}. Similarly, the second expression simplifies to

\displaystyle \langle e^{-kb/2} i e^{kb/2} , e^{i(\pi-\alpha)/2} k e^{-i(\pi-\alpha)/2}\rangle,

which by (2), (3) simplifies to {\sin b \sin \alpha}. Equating the two and rearranging, we obtain

\displaystyle \frac{\sin \alpha}{\sin a} = \frac{\sin \beta}{\sin b}

which is the spherical sine rule. Again, in the infinitesimal limit we obtain the familiar Euclidean sine rule

\displaystyle \frac{\sin \alpha}{a} = \frac{\sin \beta}{b}.

As a variant of the above analysis, we have from (5) again that

\displaystyle \langle e^{i(\pi-\beta)/2} e^{ka/2} e^{i(\pi-\gamma)/2} i e^{-i(\pi-\gamma)/2} e^{-ka/2} e^{-i(\pi-\beta)/2}, j \rangle

is equal to

\displaystyle \langle e^{-kc/2} e^{-i(\pi-\alpha)/2} e^{-kb/2} i e^{kb/2} e^{i(\pi-\alpha)/2} e^{kc/2}, j \rangle.

As before, the first expression simplifies to

\displaystyle \langle e^{ka/2} i e^{-ka/2}, e^{-i(\pi-\beta)/2} j e^{i(\pi-\beta)/2} \rangle

which equals {\sin a \cos \beta}. Meanwhile, the second expression can be rearranged as

\displaystyle \langle e^{-i(\pi-\alpha)/2} e^{-kb/2} i e^{kb/2} e^{i(\pi-\alpha)/2}, e^{kc/2} j e^{-kc/2} \rangle.

By (2), (3) we can simplify

\displaystyle e^{-i(\pi-\alpha)/2} e^{-kb/2} i e^{kb/2} e^{i(\pi-\alpha)/2}

\displaystyle = (\cos b) i - (\sin b) \cos(\pi-\alpha) j + (\sin b) \sin(\pi-\alpha) k

and so the inner product is {\cos b \sin c - \cos b \sin c \cos \alpha}, leading to the “five part rule”

\displaystyle \cos b \sin c - \sin b \cos c \cos \alpha = \sin a \cos \beta.

In the case of a right-angled triangle {\beta=\pi/2}, this simplifies to one of Napier’s rules

\displaystyle \cos \alpha = \frac{\tan c}{\tan b}, \ \ \ \ \ (6)

 which in the infinitesimal limit is the familiar {\cos \alpha = \frac{c}{b}}. The other rules of Napier can be derived in a similar fashion.

Example 3 One application of Napier’s rule (6) is to determine the sunrise equation for when the sun rises and sets at a given location on the Earth, and a given time of year. For sake of argument let us work in summer, in which the declination {\delta} of the Sun is positive (due to axial tilt, it reaches a maximum of {23.5^\circ} at the summer solstice). Then the Sun subtends an angle of {\pi/2-\delta} from the pole star (Polaris in the northern hemisphere, Sigma Octantis in the southern hemisphere), and appears to rotate around that pole star once every {24} hours. On the other hand, if one is at a latitude {\phi}, then the pole star an elevation of {\phi} above the horizon. At extremely high latitudes {\phi > \pi/2-\delta}, the sun will never set (a phenomenon known as “midnight sun“); but in all other cases, at sunrise or sunset, the sun, pole star, and horizon point below the pole star will form a right-angled spherical triangle, with hypotenuse subtending an angle {\pi/2-\delta} and vertical side subtending an angle {\phi}. The angle subtended by the pole star in this triangle is {\pi-\omega}, where {\omega} is the solar hour angle {\omega} – the angle that the sun deviates from its noon position. Equation (6) then gives the sunrise equation

\displaystyle \cos(\pi-\omega) = \frac{\tan \phi}{\tan(\pi/2-\delta)}

or equivalently

\displaystyle \cos \omega = - \tan \phi \tan \delta.

A similar rule determines the time of sunset. In particular, the number of daylight hours in summer (assuming one is not in the midnight sun scenario {\phi > \pi/2 -\delta}) is given by

\displaystyle 24 - \frac{24}{\pi} \mathrm{arccos}(\tan \phi \tan \delta).

The situation in winter is similar, except that {\delta} is now negative, and polar night (no sunrise) occurs when {\phi > \pi/2+\delta}.

On the distribution of eigenvalues of GUE and its minors at fixed index

17 December, 2024 in math.PR, paper | Tags: , , , | by | 7 comments

I’ve just uploaded to the arXiv the paper “On the distribution of eigenvalues of GUE and its minors at fixed index“. This is a somewhat technical paper establishing some estimates regarding one of the most well-studied random matrix models, the Gaussian Unitary Ensemble (GUE), that were not previously in the literature, but which will be needed for some forthcoming work of Hariharan Narayanan on the limiting behavior of “hives” with GUE boundary conditions (building upon our previous joint work with Sheffield).

For sake of discussion we normalize the GUE model to be the random {N \times N} Hermitian matrix {H} whose probability density function is proportional to {e^{-\mathrm{tr} H^2}}. With this normalization, the famous Wigner semicircle law will tell us that the eigenvalues {\lambda_1 \leq \dots \leq \lambda_N} of this matrix will almost all lie in the interval {[-\sqrt{2N}, \sqrt{2N}]}, and after dividing by {\sqrt{2N}}, will asymptotically be distributed according to the semicircle distribution

\displaystyle \rho_{\mathrm{sc}}(x) := \frac{2}{\pi} (1-x^2)_+^{1/2}.

In particular, the normalized {i^{th}} eigenvalue {\lambda_i/\sqrt{2N}} should be close to the classical location {\gamma_{i/N}}, where {\gamma_{i/N}} is the unique element of {[-1,1]} such that

\displaystyle \int_{-\infty}^{\gamma_{i/N}} \rho_{\mathrm{sc}}(x)\ dx = \frac{i}{N}.

Eigenvalues can be described by their index {i} or by their (normalized) energy {\lambda_i/\sqrt{2N}}. In principle, the two descriptions are related by the classical map {i \mapsto \gamma_{i/N}} defined above, but there are microscopic fluctuations from the classical location that create subtle technical difficulties between “fixed index” results in which one focuses on a single index {i} (and neighboring indices {i+1, i-1}, etc.), and “fixed energy” results in which one focuses on a single energy {x} (and eigenvalues near this energy). The phenomenon of eigenvalue rigidity does give some control on these fluctuations, allowing one to relate “averaged index” results (in which the index {i} ranges over a mesoscopic range) with “averaged energy” results (in which the energy {x} is similarly averaged over a mesoscopic interval), but there are technical issues in passing back from averaged control to pointwise control, either for the index or energy.

We will be mostly concerned in the bulk region where the index {i} is in an inteval of the form {[\delta n, (1-\delta)n]} for some fixed {\delta>0}, or equivalently the energy {x} is in {[-1+c, 1-c]} for some fixed {c > 0}. In this region it is natural to introduce the normalized eigenvalue gaps

\displaystyle g_i := \sqrt{N/2} \rho_{\mathrm{sc}}(\gamma_{i/N}) (\lambda_{i+1} - \lambda_i).

The semicircle law predicts that these gaps {g_i} have mean close to {1}; however, due to the aforementioned fluctuations around the classical location, this type of claim is only easy to establish in the “fixed energy”, “averaged energy”, or “averaged index” settings; the “fixed index” case was only achieved by myself as recently as 2013, where I showed that each such gap in fact asymptotically had the expected distribution of the Gaudin law, using manipulations of determinantal processes. A significantly more general result, avoiding the use of determinantal processes, was subsequently obtained by Erdos and Yau.

However, these results left open the possibility of bad tail behavior at extremely large or small values of the gaps {g_i}; in particular, moments of the {g_i} were not directly controlled by previous results. The first result of the paper is to push the determinantal analysis further, and obtain such results. For instance, we obtain moment bounds

\displaystyle \mathop{\bf E} g_i^p \ll_p 1

for any fixed {p > 0}, as well as an exponential decay bound

\displaystyle \mathop{\bf P} (g_i > h) \ll \exp(-h/4)

for {0 < h \ll \log\log N}, and a lower tail bound

\displaystyle \mathop{\bf P} (g_i \leq h) \ll h^{2/3} \log^{1/2} \frac{1}{h}

for any {h>0}. We also obtain good control on sums {g_i + \dots + g_{i+m-1}} of {m} consecutive gaps for any fixed {m}, showing that this sum has mean {m + O(\log^{4/3} (2+m))} and variance {O(\log^{7/3} (2+m))}. (This is significantly less variance than one would expect from a sum of {m} independent random variables; this variance reduction phenomenon is closely related to the eigenvalue rigidity phenomenon alluded to earlier, and reflects the tendency of eigenvalues to repel each other.)

A key point in these estimates is that no factors of {\log N} occur in the estimates, which is what one would obtain if one tried to use existing eigenvalue rigidity theorems. (In particular, if one normalized the eigenvalues {\lambda_i} at the same scale at the gap {g_i}, they would fluctuate by a standard deviation of about {\sqrt{\log N}}; it is only the gaps between eigenvalues that exhibit much smaller fluctuation.) On the other hand, the dependence on {h} is not optimal, although it was sufficient for the applications I had in mind.

As with my previous paper, the strategy is to try to replace fixed index events such as {g_i > h} with averaged energy events. For instance, if {g_i > h} and {i} has classical location {x}, then there is an interval of normalized energies {t} of length {\gg h}, with the property that there are precisely {N-i} eigenvalues to the right of {f_x(t)} and no eigenvalues in the interval {[f_x(t), f_x(t+h/2)]}, where

\displaystyle f_x(t) = \sqrt{2N}( x + \frac{t}{N \rho_{\mathrm{sc}}(x)})

is an affine rescaling to the scale of the eigenvalue gap. So matters soon reduce to controlling the probability of the event

\displaystyle (N_{x,t} = N-i) \wedge (N_{x,t,h/2} = 0)

where {N_{x,t}} is the number of eigenvalues to the right of {f_x(t)}, and {N_{x,t,h/2}} is the number of eigenvalues in the interval {[f_x(t), f_x(t+h/2)]}. These are fixed energy events, and one can use the theory of determinantal processes to control them. For instance, each of the random variables {N_{x,t}}, {N_{x,t,h/2}} separately have the distribution of sums of independent Boolean variables, which are extremely well understood. Unfortunately, the coupling is a problem; conditioning on the event {N_{x,t} = N-i}, in particular, affects the distribution of {N_{x,t,h/2}}, so that it is no longer the sum of independent Boolean variables. However, it is still a mixture of such sums, and with this (and the Plancherel-Rotach asymptotics for the GUE determinantal kernel) it is possible to proceed and obtain the above estimates after some calculation.

For the intended application to GUE hives, it is important to not just control gaps {g_i} of the eigenvalues {\lambda_i} of the GUE matrix {M}, but also the gaps {g'_i} of the eigenvalues {\lambda'_i} of the top left {N-1 \times N-1} minor {M'} of {M}. This minor of a GUE matrix is basically again a GUE matrix, so the above theorem applies verbatim to the {g'_i}; but it turns out to be necessary to control the joint distribution of the {g_i} and {g'_i}, and also of the interlacing gaps {\tilde g_i} between the {\lambda_i} and {\lambda'_i}. For fixed energy, these gaps are in principle well understood, due to previous work of Adler-Nordenstam-van Moerbeke and of Johansson-Nordenstam which show that the spectrum of both matrices is asymptotically controlled by the Boutillier bead process. This also gives averaged energy and averaged index results without much difficulty, but to get to fixed index information, one needs some universality result in the index {i}. For the gaps {g_i} of the original matrix, such a universality result is available due to the aforementioned work of Erdos and Yau, but this does not immediately imply the corresponding universality result for the joint distribution of {g_i} and {g'_i} or {\tilde g_i}. For this, we need a way to relate the eigenvalues {\lambda_i} of the matrix {M} to the eigenvalues {\lambda'_i} of the minors {M'}. By a standard Schur’s complement calculation, one can obtain the equation

\displaystyle a_{NN} - \lambda_i - \sum_{j=1}^{N-1}\frac{|X_j|^2}{\lambda'_j - \lambda_i} = 0

for all {i}, where {a_{NN}} is the bottom right entry of {M}, and {X_1,\dots,X_{N-1}} are complex gaussians independent of {\lambda'_j}. This gives a random system of equations to solve for {\lambda_i} in terms of {\lambda'_j}. Using the previous bounds on eigenvalue gaps (particularly the concentration results for sums of consecutive gaps), one can localize this equation to the point where a given {\lambda_i} is mostly controlled by a bounded number of nearby {\lambda'_j}, and hence a single gap {g_i} is mostly controlled by a bounded number of {g'_j}. From this, it is possible to leverage the existing universality result of Erdos and Yau to obtain universality of the joint distribution of {g_i} and {g'_i} (or of {\tilde g_i}). (The result can also be extended to more layers of the minor process than just two, as long as the number of minors is held fixed.)

This at last brings us to the final result of the paper, which is the one which is actually needed for the application to GUE hives. Here, one is interested in controlling the variance of a linear combination {\sum_{l=1}^m a_l \tilde g_{i+l}} of a fixed number {l} of consecutive interlacing gaps {\tilde g_{i+l}}, where the {a_l} are arbitrary deterministic coefficients. An application of the triangle and Cauchy-Schwarz inequalities, combined with the previous moment bounds on gaps, shows that this randomv ariable has variance {\ll m \sum_{l=1}^m |a_i|^2}. However, this bound is not expected to be sharp, due to the expected decay between correlations of eigenvalue gaps. In this paper, I improve the variance bound to

\displaystyle \ll_A \frac{m}{\log^A(2+m)} \sum_{l=1}^m |a_i|^2

for any {A>0}, which is what was needed for the application.

This improvement reflects some decay in the covariances between distant interlacing gaps {\tilde g_i, \tilde g_{i+h}}. I was not able to establish such decay directly. Instead, using some Fourier analysis, one can reduce matters to studying the case of modulated linear statistics such as {\sum_{l=1}^m e(\xi l) \tilde g_{i+l}} for various frequencies {\xi}. In “high frequency” cases one can use the triangle inequality to reduce matters to studying the original eigenvalue gaps {g_i}, which can be handled by a (somewhat complicated) determinantal process calculation, after first using universality results to pass from fixed index to averaged index, thence to averaged energy, then to fixed energy estimates. For low frequencies the triangle inequality argument is unfavorable, and one has to instead use the determinantal kernel of the full minor process, and not just an individual matrix. This requires some classical, but tedious, calculation of certain asymptotics of sums involving Hermite polynomials.

The full argument is unfortunately quite complex, but it seems that the combination of having to deal with minors, as well as fixed indices, places this result out of reach of many prior methods.

AI for Math fund

5 December, 2024 in advertising | Tags: | by | 20 comments

Renaissance Philanthropy and XTX Markets have announced the launch of the AI for Math Fund, a new grant program supporting projects that apply AI and machine learning to mathematics, with a focus on automated theorem proving, with an initial $9.2 million in funding. The project funding categories, and examples of projects in such categories, are:

1. Production Grade Software Tools

  • AI-based autoformalization tools for translating natural-language mathematics into the formalisms of proof assistants
  • AI-based auto-informalization tools for translating proof-assistant proofs into interpretable natural-language mathematics
  • AI-based models for suggesting tactics/steps or relevant concepts to the user of a proof assistant, or for generating entire proofs
  • Infrastructure to connect proof assistants with computer algebra systems, calculus, and PDEs
  • A large-scale, AI-enhanced distributed collaboration platform for mathematicians

2. Datasets

  • Datasets of formalized theorems and proofs in a proof assistant
  • Datasets that would advance AI for theorem proving as applied to program verification and secure code generation
  • Datasets of (natural-language) mathematical problems, theorems, proofs, exposition, etc.
  • Benchmarks and training environments associated with datasets and model tasks (autoformalization, premise selection, tactic or proof generation, etc.)

3. Field Building

  • Textbooks
  • Courses
  • Documentation and support for proof assistants, and interfaces/APIs to integrate with AI tools

4. Breakthrough Ideas

  • Expected difficulty estimation (of sub-problems of a proof)
  • Novel mathematical implications of proofs formalized type-theoretically
  • Formalization of proof complexity in proof assistants

The deadline for initial expressions of interest is Jan 10, 2025.

[Disclosure: I have agreed to serve on the advisory board for this fund.]

Update: See also this discussion thread on possible projects that might be supported by this fund.

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