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Dimitri Shlyakhtenko and I have uploaded to the arXiv our paper Fractional free convolution powers. For me, this project (which we started during the 2018 IPAM program on quantitative linear algebra) was motivated by a desire to understand the behavior of the minor process applied to a large random Hermitian ${N \times N}$ matrix ${A_N}$, in which one takes the successive upper left ${n \times n}$ minors ${A_n}$ of ${A_N}$ and computes their eigenvalues ${\lambda_1(A_n) \leq \dots \leq \lambda_n(A_n)}$ in non-decreasing order. These eigenvalues are related to each other by the Cauchy interlacing inequalities

$\displaystyle \lambda_i(A_{n+1}) \leq \lambda_i(A_n) \leq \lambda_{i+1}(A_{n+1})$

for ${1 \leq i \leq n < N}$, and are often arranged in a triangular array known as a Gelfand-Tsetlin pattern, as discussed in these previous blog posts.

When ${N}$ is large and the matrix ${A_N}$ is a random matrix with empirical spectral distribution converging to some compactly supported probability measure ${\mu}$ on the real line, then under suitable hypotheses (e.g., unitary conjugation invariance of the random matrix ensemble ${A_N}$), a “concentration of measure” effect occurs, with the spectral distribution of the minors ${A_n}$ for ${n = \lfloor N/k\rfloor}$ for any fixed ${k \geq 1}$ converging to a specific measure ${k^{-1}_* \mu^{\boxplus k}}$ that depends only on ${\mu}$ and ${k}$. The reason for this notation is that there is a surprising description of this measure ${k^{-1}_* \mu^{\boxplus k}}$ when ${k}$ is a natural number, namely it is the free convolution ${\mu^{\boxplus k}}$ of ${k}$ copies of ${\mu}$, pushed forward by the dilation map ${x \mapsto k^{-1} x}$. For instance, if ${\mu}$ is the Wigner semicircular measure ${d\mu_{sc} = \frac{1}{\pi} (4-x^2)^{1/2}_+\ dx}$, then ${k^{-1}_* \mu_{sc}^{\boxplus k} = k^{-1/2}_* \mu_{sc}}$. At the random matrix level, this reflects the fact that the minor of a GUE matrix is again a GUE matrix (up to a renormalizing constant).

As first observed by Bercovici and Voiculescu and developed further by Nica and Speicher, among other authors, the notion of a free convolution power ${\mu^{\boxplus k}}$ of ${\mu}$ can be extended to non-integer ${k \geq 1}$, thus giving the notion of a “fractional free convolution power”. This notion can be defined in several different ways. One of them proceeds via the Cauchy transform

$\displaystyle G_\mu(z) := \int_{\bf R} \frac{d\mu(x)}{z-x}$

of the measure ${\mu}$, and ${\mu^{\boxplus k}}$ can be defined by solving the Burgers-type equation

$\displaystyle (k \partial_k + z \partial_z) G_{\mu^{\boxplus k}}(z) = \frac{\partial_z G_{\mu^{\boxplus k}}(z)}{G_{\mu^{\boxplus k}}(z)} \ \ \ \ \ (1)$

with initial condition ${G_{\mu^{\boxplus 1}} = G_\mu}$ (see this previous blog post for a derivation). This equation can be solved explicitly using the ${R}$-transform ${R_\mu}$ of ${\mu}$, defined by solving the equation

$\displaystyle \frac{1}{G_\mu(z)} + R_\mu(G_\mu(z)) = z$

for sufficiently large ${z}$, in which case one can show that

$\displaystyle R_{\mu^{\boxplus k}}(z) = k R_\mu(z).$

(In the case of the semicircular measure ${\mu_{sc}}$, the ${R}$-transform is simply the identity: ${R_{\mu_{sc}}(z)=z}$.)

Nica and Speicher also gave a free probability interpretation of the fractional free convolution power: if ${A}$ is a noncommutative random variable in a noncommutative probability space ${({\mathcal A},\tau)}$ with distribution ${\mu}$, and ${p}$ is a real projection operator free of ${A}$ with trace ${1/k}$, then the “minor” ${[pAp]}$ of ${A}$ (viewed as an element of a new noncommutative probability space ${({\mathcal A}_p, \tau_p)}$ whose elements are minors ${[pXp]}$, ${X \in {\mathcal A}}$ with trace ${\tau_p([pXp]) := k \tau(pXp)}$) has the law of ${k^{-1}_* \mu^{\boxplus k}}$ (we give a self-contained proof of this in an appendix to our paper). This suggests that the minor process (or fractional free convolution) can be studied within the framework of free probability theory.

One of the known facts about integer free convolution powers ${\mu^{\boxplus k}}$ is monotonicity of the free entropy

$\displaystyle \chi(\mu) = \int_{\bf R} \int_{\bf R} \log|s-t|\ d\mu(s) d\mu(t) + \frac{3}{4} + \frac{1}{2} \log 2\pi$

and free Fisher information

$\displaystyle \Phi(\mu) = \frac{2\pi^2}{3} \int_{\bf R} \left(\frac{d\mu}{dx}\right)^3\ dx$

which were introduced by Voiculescu as free probability analogues of the classical probability concepts of differential entropy and classical Fisher information. (Here we correct a small typo in the normalization constant of Fisher entropy as presented in Voiculescu’s paper.) Namely, it was shown by Shylakhtenko that the quantity ${\chi(k^{-1/2}_* \mu^{\boxplus k})}$ is monotone non-decreasing for integer ${k}$, and the Fisher information ${\Phi(k^{-1/2}_* \mu^{\boxplus k})}$ is monotone non-increasing for integer ${k}$. This is the free probability analogue of the corresponding monotonicities for differential entropy and classical Fisher information that was established by Artstein, Ball, Barthe, and Naor, answering a question of Shannon.

Our first main result is to extend the monotonicity results of Shylakhtenko to fractional ${k \geq 1}$. We give two proofs of this fact, one using free probability machinery, and a more self contained (but less motivated) proof using integration by parts and contour integration. The free probability proof relies on the concept of the free score ${J(X)}$ of a noncommutative random variable, which is the analogue of the classical score. The free score, also introduced by Voiculescu, can be defined by duality as measuring the perturbation with respect to semicircular noise, or more precisely

$\displaystyle \frac{d}{d\varepsilon} \tau( Z P( X + \varepsilon Z) )|_{\varepsilon=0} = \tau( J(X) P(X) )$

whenever ${P}$ is a polynomial and ${Z}$ is a semicircular element free of ${X}$. If ${X}$ has an absolutely continuous law ${\mu = f\ dx}$ for a sufficiently regular ${f}$, one can calculate ${J(X)}$ explicitly as ${J(X) = 2\pi Hf(X)}$, where ${Hf}$ is the Hilbert transform of ${f}$, and the Fisher information is given by the formula

$\displaystyle \Phi(X) = \tau( J(X)^2 ).$

One can also define a notion of relative free score ${J(X:B)}$ relative to some subalgebra ${B}$ of noncommutative random variables.

The free score interacts very well with the free minor process ${X \mapsto [pXp]}$, in particular by standard calculations one can establish the identity

$\displaystyle J( [pXp] : [pBp] ) = k {\bf E}( [p J(X:B) p] | [pXp], [pBp] )$

whenever ${X}$ is a noncommutative random variable, ${B}$ is an algebra of noncommutative random variables, and ${p}$ is a real projection of trace ${1/k}$ that is free of both ${X}$ and ${B}$. The monotonicity of free Fisher information then follows from an application of Pythagoras’s theorem (which implies in particular that conditional expectation operators are contractions on ${L^2}$). The monotonicity of free entropy then follows from an integral representation of free entropy as an integral of free Fisher information along the free Ornstein-Uhlenbeck process (or equivalently, free Fisher information is essentially the rate of change of free entropy with respect to perturbation by semicircular noise). The argument also shows when equality holds in the monotonicity inequalities; this occurs precisely when ${\mu}$ is a semicircular measure up to affine rescaling.

After an extensive amount of calculation of all the quantities that were implicit in the above free probability argument (in particular computing the various terms involved in the application of Pythagoras’ theorem), we were able to extract a self-contained proof of monotonicity that relied on differentiating the quantities in ${k}$ and using the differential equation (1). It turns out that if ${d\mu = f\ dx}$ for sufficiently regular ${f}$, then there is an identity

$\displaystyle \partial_k \Phi( k^{-1/2}_* \mu^{\boxplus k} ) = -\frac{1}{2\pi^2} \lim_{\varepsilon \rightarrow 0} \sum_{\alpha,\beta = \pm} f(x) f(y) K(x+i\alpha \varepsilon, y+i\beta \varepsilon)\ dx dy \ \ \ \ \ (2)$

where ${K}$ is the kernel

$\displaystyle K(z,w) := \frac{1}{G(z) G(w)} (\frac{G(z)-G(w)}{z-w} + G(z) G(w))^2$

and ${G(z) := G_\mu(z)}$. It is not difficult to show that ${K(z,\overline{w})}$ is a positive semi-definite kernel, which gives the required monotonicity. It would be interesting to obtain some more insightful interpretation of the kernel ${K}$ and the identity (2).

These monotonicity properties hint at the minor process ${A \mapsto [pAp]}$ being associated to some sort of “gradient flow” in the ${k}$ parameter. We were not able to formalize this intuition; indeed, it is not clear what a gradient flow on a varying noncommutative probability space ${({\mathcal A}_p, \tau_p)}$ even means. However, after substantial further calculation we were able to formally describe the minor process as the Euler-Lagrange equation for an intriguing Lagrangian functional that we conjecture to have a random matrix interpretation. We first work in “Lagrangian coordinates”, defining the quantity ${\lambda(s,y)}$ on the “Gelfand-Tsetlin pyramid”

$\displaystyle \Delta = \{ (s,y): 0 < s < 1; 0 < y < s \}$

by the formula

$\displaystyle \mu^{\boxplus 1/s}((-\infty,\lambda(s,y)/s])=y/s,$

which is well defined if the density of ${\mu}$ is sufficiently well behaved. The random matrix interpretation of ${\lambda(s,y)}$ is that it is the asymptotic location of the ${\lfloor yN\rfloor^{th}}$ eigenvalue of the ${\lfloor sN \rfloor \times \lfloor sN \rfloor}$ upper left minor of a random ${N \times N}$ matrix ${A_N}$ with asymptotic empirical spectral distribution ${\mu}$ and with unitarily invariant distribution, thus ${\lambda}$ is in some sense a continuum limit of Gelfand-Tsetlin patterns. Thus for instance the Cauchy interlacing laws in this asymptotic limit regime become

$\displaystyle 0 \leq \partial_s \lambda \leq \partial_y \lambda.$

After a lengthy calculation (involving extensive use of the chain rule and product rule), the equation (1) is equivalent to the Euler-Lagrange equation

$\displaystyle \partial_s L_{\lambda_s}(\partial_s \lambda, \partial_y \lambda) + \partial_y L_{\lambda_y}(\partial_s \lambda, \partial_y \lambda) = 0$

where ${L}$ is the Lagrangian density

$\displaystyle L(\lambda_s, \lambda_y) := \log \lambda_y + \log \sin( \pi \frac{\lambda_s}{\lambda_y} ).$

Thus the minor process is formally a critical point of the integral ${\int_\Delta L(\partial_s \lambda, \partial_y \lambda)\ ds dy}$. The quantity ${\partial_y \lambda}$ measures the mean eigenvalue spacing at some location of the Gelfand-Tsetlin pyramid, and the ratio ${\frac{\partial_s \lambda}{\partial_y \lambda}}$ measures mean eigenvalue drift in the minor process. This suggests that this Lagrangian density is some sort of measure of entropy of the asymptotic microscale point process emerging from the minor process at this spacing and drift. There is work of Metcalfe demonstrating that this point process is given by the Boutillier bead model, so we conjecture that this Lagrangian density ${L}$ somehow measures the entropy density of this process.

Suppose we have an ${n \times n}$ matrix ${M}$ that is expressed in block-matrix form as

$\displaystyle M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$

where ${A}$ is an ${(n-k) \times (n-k)}$ matrix, ${B}$ is an ${(n-k) \times k}$ matrix, ${C}$ is an ${k \times (n-k)}$ matrix, and ${D}$ is a ${k \times k}$ matrix for some ${1 < k < n}$. If ${A}$ is invertible, we can use the technique of Schur complementation to express the inverse of ${M}$ (if it exists) in terms of the inverse of ${A}$, and the other components ${B,C,D}$ of course. Indeed, to solve the equation

$\displaystyle M \begin{pmatrix} x & y \end{pmatrix} = \begin{pmatrix} a & b \end{pmatrix},$

where ${x, a}$ are ${(n-k) \times 1}$ column vectors and ${y,b}$ are ${k \times 1}$ column vectors, we can expand this out as a system

$\displaystyle Ax + By = a$

$\displaystyle Cx + Dy = b.$

Using the invertibility of ${A}$, we can write the first equation as

$\displaystyle x = A^{-1} a - A^{-1} B y \ \ \ \ \ (1)$

and substituting this into the second equation yields

$\displaystyle (D - C A^{-1} B) y = b - C A^{-1} a$

and thus (assuming that ${D - CA^{-1} B}$ is invertible)

$\displaystyle y = - (D - CA^{-1} B)^{-1} CA^{-1} a + (D - CA^{-1} B)^{-1} b$

and then inserting this back into (1) gives

$\displaystyle x = (A^{-1} + A^{-1} B (D - CA^{-1} B)^{-1} C A^{-1}) a - A^{-1} B (D - CA^{-1} B)^{-1} b.$

Comparing this with

$\displaystyle \begin{pmatrix} x & y \end{pmatrix} = M^{-1} \begin{pmatrix} a & b \end{pmatrix},$

we have managed to express the inverse of ${M}$ as

$\displaystyle M^{-1} =$

$\displaystyle \begin{pmatrix} A^{-1} + A^{-1} B (D - CA^{-1} B)^{-1} C A^{-1} & - A^{-1} B (D - CA^{-1} B)^{-1} \\ - (D - CA^{-1} B)^{-1} CA^{-1} & (D - CA^{-1} B)^{-1} \end{pmatrix}. \ \ \ \ \ (2)$

One can consider the inverse problem: given the inverse ${M^{-1}}$ of ${M}$, does one have a nice formula for the inverse ${A^{-1}}$ of the minor ${A}$? Trying to recover this directly from (2) looks somewhat messy. However, one can proceed as follows. Let ${U}$ denote the ${n \times k}$ matrix

$\displaystyle U := \begin{pmatrix} 0 \\ I_k \end{pmatrix}$

(with ${I_k}$ the ${k \times k}$ identity matrix), and let ${V}$ be its transpose:

$\displaystyle V := \begin{pmatrix} 0 & I_k \end{pmatrix}.$

Then for any scalar ${t}$ (which we identify with ${t}$ times the identity matrix), one has

$\displaystyle M + UtV = \begin{pmatrix} A & B \\ C & D+t \end{pmatrix},$

and hence by (2)

$\displaystyle (M+UtV)^{-1} =$

$\displaystyle \begin{pmatrix} A^{-1} + A^{-1} B (D + t - CA^{-1} B)^{-1} C A^{-1} & - A^{-1} B (D + t- CA^{-1} B)^{-1} \\ - (D + t - CA^{-1} B)^{-1} CA^{-1} & (D + t - CA^{-1} B)^{-1} \end{pmatrix}.$

noting that the inverses here will exist for ${t}$ large enough. Taking limits as ${t \rightarrow \infty}$, we conclude that

$\displaystyle \lim_{t \rightarrow \infty} (M+UtV)^{-1} = \begin{pmatrix} A^{-1} & 0 \\ 0 & 0 \end{pmatrix}.$

On the other hand, by the Woodbury matrix identity (discussed in this previous blog post), we have

$\displaystyle (M+UtV)^{-1} = M^{-1} - M^{-1} U (t^{-1} + V M^{-1} U)^{-1} V M^{-1}$

and hence on taking limits and comparing with the preceding identity, one has

$\displaystyle \begin{pmatrix} A^{-1} & 0 \\ 0 & 0 \end{pmatrix} = M^{-1} - M^{-1} U (V M^{-1} U)^{-1} V M^{-1}.$

This achieves the aim of expressing the inverse ${A^{-1}}$ of the minor in terms of the inverse of the full matrix. Taking traces and rearranging, we conclude in particular that

$\displaystyle \mathrm{tr} A^{-1} = \mathrm{tr} M^{-1} - \mathrm{tr} (V M^{-2} U) (V M^{-1} U)^{-1}. \ \ \ \ \ (3)$

In the ${k=1}$ case, this can be simplified to

$\displaystyle \mathrm{tr} A^{-1} = \mathrm{tr} M^{-1} - \frac{e_n^T M^{-2} e_n}{e_n^T M^{-1} e_n} \ \ \ \ \ (4)$

where ${e_n}$ is the ${n^{th}}$ basis column vector.

We can apply this identity to understand how the spectrum of an ${n \times n}$ random matrix ${M}$ relates to that of its top left ${n-1 \times n-1}$ minor ${A}$. Subtracting any complex multiple ${z}$ of the identity from ${M}$ (and hence from ${A}$), we can relate the Stieltjes transform ${s_M(z) := \frac{1}{n} \mathrm{tr}(M-z)^{-1}}$ of ${M}$ with the Stieltjes transform ${s_A(z) := \frac{1}{n-1} \mathrm{tr}(A-z)^{-1}}$ of ${A}$:

$\displaystyle s_A(z) = \frac{n}{n-1} s_M(z) - \frac{1}{n-1} \frac{e_n^T (M-z)^{-2} e_n}{e_n^T (M-z)^{-1} e_n} \ \ \ \ \ (5)$

At this point we begin to proceed informally. Assume for sake of argument that the random matrix ${M}$ is Hermitian, with distribution that is invariant under conjugation by the unitary group ${U(n)}$; for instance, ${M}$ could be drawn from the Gaussian Unitary Ensemble (GUE), or alternatively ${M}$ could be of the form ${M = U D U^*}$ for some real diagonal matrix ${D}$ and ${U}$ a unitary matrix drawn randomly from ${U(n)}$ using Haar measure. To fix normalisations we will assume that the eigenvalues of ${M}$ are typically of size ${O(1)}$. Then ${A}$ is also Hermitian and ${U(n)}$-invariant. Furthermore, the law of ${e_n^T (M-z)^{-1} e_n}$ will be the same as the law of ${u^* (M-z)^{-1} u}$, where ${u}$ is now drawn uniformly from the unit sphere (independently of ${M}$). Diagonalising ${M}$ into eigenvalues ${\lambda_j}$ and eigenvectors ${v_j}$, we have

$\displaystyle u^* (M-z)^{-1} u = \sum_{j=1}^n \frac{|u^* v_j|^2}{\lambda_j - z}.$

One can think of ${u}$ as a random (complex) Gaussian vector, divided by the magnitude of that vector (which, by the Chernoff inequality, will concentrate to ${\sqrt{n}}$). Thus the coefficients ${u^* v_j}$ with respect to the orthonormal basis ${v_1,\dots,v_j}$ can be thought of as independent (complex) Gaussian vectors, divided by that magnitude. Using this and the Chernoff inequality again, we see (for ${z}$ distance ${\sim 1}$ away from the real axis at least) that one has the concentration of measure

$\displaystyle u^* (M-z)^{-1} u \approx \frac{1}{n} \sum_{j=1}^n \frac{1}{\lambda_j - z}$

and thus

$\displaystyle e_n^T (M-z)^{-1} e_n \approx \frac{1}{n} \mathrm{tr} (M-z)^{-1} = s_M(z)$

(that is to say, the diagonal entries of ${(M-z)^{-1}}$ are roughly constant). Similarly we have

$\displaystyle e_n^T (M-z)^{-2} e_n \approx \frac{1}{n} \mathrm{tr} (M-z)^{-2} = \frac{d}{dz} s_M(z).$

Inserting this into (5) and discarding terms of size ${O(1/n^2)}$, we thus conclude the approximate relationship

$\displaystyle s_A(z) \approx s_M(z) + \frac{1}{n} ( s_M(z) - s_M(z)^{-1} \frac{d}{dz} s_M(z) ).$

This can be viewed as a difference equation for the Stieltjes transform of top left minors of ${M}$. Iterating this equation, and formally replacing the difference equation by a differential equation in the large ${n}$ limit, we see that when ${n}$ is large and ${k \approx e^{-t} n}$ for some ${t \geq 0}$, one expects the top left ${k \times k}$ minor ${A_k}$ of ${M}$ to have Stieltjes transform

$\displaystyle s_{A_k}(z) \approx s( t, z ) \ \ \ \ \ (6)$

where ${s(t,z)}$ solves the Burgers-type equation

$\displaystyle \partial_t s(t,z) = s(t,z) - s(t,z)^{-1} \frac{d}{dz} s(t,z) \ \ \ \ \ (7)$

with initial data ${s(0,z) = s_M(z)}$.

Example 1 If ${M}$ is a constant multiple ${M = cI_n}$ of the identity, then ${s_M(z) = \frac{1}{c-z}}$. One checks that ${s(t,z) = \frac{1}{c-z}}$ is a steady state solution to (7), which is unsurprising given that all minors of ${M}$ are also ${c}$ times the identity.

Example 2 If ${M}$ is GUE normalised so that each entry has variance ${\sigma^2/n}$, then by the semi-circular law (see previous notes) one has ${s_M(z) \approx \frac{-z + \sqrt{z^2-4\sigma^2}}{2\sigma^2} = -\frac{2}{z + \sqrt{z^2-4\sigma^2}}}$ (using an appropriate branch of the square root). One can then verify the self-similar solution

$\displaystyle s(t,z) = \frac{-z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}{2\sigma^2 e^{-t}} = -\frac{2}{z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}$

to (7), which is consistent with the fact that a top ${k \times k}$ minor of ${M}$ also has the law of GUE, with each entry having variance ${\sigma^2 / n \approx \sigma^2 e^{-t} / k}$ when ${k \approx e^{-t} n}$.

One can justify the approximation (6) given a sufficiently good well-posedness theory for the equation (7). We will not do so here, but will note that (as with the classical inviscid Burgers equation) the equation can be solved exactly (formally, at least) by the method of characteristics. For any initial position ${z_0}$, we consider the characteristic flow ${t \mapsto z(t)}$ formed by solving the ODE

$\displaystyle \frac{d}{dt} z(t) = s(t,z(t))^{-1} \ \ \ \ \ (8)$

with initial data ${z(0) = z_0}$, ignoring for this discussion the problems of existence and uniqueness. Then from the chain rule, the equation (7) implies that

$\displaystyle \frac{d}{dt} s( t, z(t) ) = s(t,z(t))$

and thus ${s(t,z(t)) = e^t s(0,z_0)}$. Inserting this back into (8) we see that

$\displaystyle z(t) = z_0 + s(0,z_0)^{-1} (1-e^{-t})$

and thus (7) may be solved implicitly via the equation

$\displaystyle s(t, z_0 + s(0,z_0)^{-1} (1-e^{-t}) ) = e^t s(0, z_0) \ \ \ \ \ (9)$

for all ${t}$ and ${z_0}$.

Remark 3 In practice, the equation (9) may stop working when ${z_0 + s(0,z_0)^{-1} (1-e^{-t})}$ crosses the real axis, as (7) does not necessarily hold in this region. It is a cute exercise (ultimately coming from the Cauchy-Schwarz inequality) to show that this crossing always happens, for instance if ${z_0}$ has positive imaginary part then ${z_0 + s(0,z_0)^{-1}}$ necessarily has negative or zero imaginary part.

Example 4 Suppose we have ${s(0,z) = \frac{1}{c-z}}$ as in Example 1. Then (9) becomes

$\displaystyle s( t, z_0 + (c-z_0) (1-e^{-t}) ) = \frac{e^t}{c-z_0}$

for any ${t,z_0}$, which after making the change of variables ${z = z_0 + (c-z_0) (1-e^{-t}) = c - e^{-t} (c - z_0)}$ becomes

$\displaystyle s(t, z ) = \frac{1}{c-z}$

as in Example 1.

Example 5 Suppose we have

$\displaystyle s(0,z) = \frac{-z + \sqrt{z^2-4\sigma^2}}{2\sigma^2} = -\frac{2}{z + \sqrt{z^2-4\sigma^2}}.$

as in Example 2. Then (9) becomes

$\displaystyle s(t, z_0 - \frac{z_0 + \sqrt{z_0^2-4\sigma^2}}{2} (1-e^{-t}) ) = e^t \frac{-z_0 + \sqrt{z_0^2-4\sigma^2}}{2\sigma^2}.$

If we write

$\displaystyle z := z_0 - \frac{z_0 + \sqrt{z_0^2-4\sigma^2}}{2} (1-e^{-t})$

$\displaystyle = \frac{(1+e^{-t}) z_0 - (1-e^{-t}) \sqrt{z_0^2-4\sigma^2}}{2}$

one can calculate that

$\displaystyle z^2 - 4 \sigma^2 e^{-t} = (\frac{(1-e^{-t}) z_0 - (1+e^{-t}) \sqrt{z_0^2-4\sigma^2}}{2})^2$

and hence

$\displaystyle \frac{-z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}{2\sigma^2 e^{-t}} = e^t \frac{-z_0 + \sqrt{z_0^2-4\sigma^2}}{2\sigma^2}$

which gives

$\displaystyle s(t,z) = \frac{-z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}{2\sigma^2 e^{-t}}. \ \ \ \ \ (10)$

One can recover the spectral measure ${\mu}$ from the Stieltjes transform ${s(z)}$ as the weak limit of ${x \mapsto \frac{1}{\pi} \mathrm{Im} s(x+i\varepsilon)}$ as ${\varepsilon \rightarrow 0}$; we write this informally as

$\displaystyle d\mu(x) = \frac{1}{\pi} \mathrm{Im} s(x+i0^+)\ dx.$

In this informal notation, we have for instance that

$\displaystyle \delta_c(x) = \frac{1}{\pi} \mathrm{Im} \frac{1}{c-x-i0^+}\ dx$

which can be interpreted as the fact that the Cauchy distributions ${\frac{1}{\pi} \frac{\varepsilon}{(c-x)^2+\varepsilon^2}}$ converge weakly to the Dirac mass at ${c}$ as ${\varepsilon \rightarrow 0}$. Similarly, the spectral measure associated to (10) is the semicircular measure ${\frac{1}{2\pi \sigma^2 e^{-t}} (4 \sigma^2 e^{-t}-x^2)_+^{1/2}}$.

If we let ${\mu_t}$ be the spectral measure associated to ${s(t,\cdot)}$, then the curve ${e^{-t} \mapsto \mu_t}$ from ${(0,1]}$ to the space of measures is the high-dimensional limit ${n \rightarrow \infty}$ of a Gelfand-Tsetlin pattern (discussed in this previous post), if the pattern is randomly generated amongst all matrices ${M}$ with spectrum asymptotic to ${\mu_0}$ as ${n \rightarrow \infty}$. For instance, if ${\mu_0 = \delta_c}$, then the curve is ${\alpha \mapsto \delta_c}$, corresponding to a pattern that is entirely filled with ${c}$‘s. If instead ${\mu_0 = \frac{1}{2\pi \sigma^2} (4\sigma^2-x^2)_+^{1/2}}$ is a semicircular distribution, then the pattern is

$\displaystyle \alpha \mapsto \frac{1}{2\pi \sigma^2 \alpha} (4\sigma^2 \alpha -x^2)_+^{1/2},$

thus at height ${\alpha}$ from the top, the pattern is semicircular on the interval ${[-2\sigma \sqrt{\alpha}, 2\sigma \sqrt{\alpha}]}$. The interlacing property of Gelfand-Tsetlin patterns translates to the claim that ${\alpha \mu_\alpha(-\infty,\lambda)}$ (resp. ${\alpha \mu_\alpha(\lambda,\infty)}$) is non-decreasing (resp. non-increasing) in ${\alpha}$ for any fixed ${\lambda}$. In principle one should be able to establish these monotonicity claims directly from the PDE (7) or from the implicit solution (9), but it was not clear to me how to do so.

An interesting example of such a limiting Gelfand-Tsetlin pattern occurs when ${\mu_0 = \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1}$, which corresponds to ${M}$ being ${2P-I}$, where ${P}$ is an orthogonal projection to a random ${n/2}$-dimensional subspace of ${{\bf C}^n}$. Here we have

$\displaystyle s(0,z) = \frac{1}{2} \frac{1}{-1-z} + \frac{1}{2} \frac{1}{1-z} = \frac{z}{1-z^2}$

and so (9) in this case becomes

$\displaystyle s(t, z_0 + \frac{1-z_0^2}{z_0} (1-e^{-t}) ) = \frac{e^t z_0}{1-z_0^2}$

A tedious calculation then gives the solution

$\displaystyle s(t,z) = \frac{(2e^{-t}-1)z + \sqrt{z^2 - 4e^{-t}(1-e^{-t})}}{2e^{-t}(1-z^2)}. \ \ \ \ \ (11)$

For ${\alpha = e^{-t} > 1/2}$, there are simple poles at ${z=-1,+1}$, and the associated measure is

$\displaystyle \mu_\alpha = \frac{2\alpha-1}{2\alpha} \delta_{-1} + \frac{2\alpha-1}{2\alpha} \delta_1 + \frac{1}{2\pi \alpha(1-x^2)} (4\alpha(1-\alpha)-x^2)_+^{1/2}\ dx.$

This reflects the interlacing property, which forces ${\frac{2\alpha-1}{2\alpha} \alpha n}$ of the ${\alpha n}$ eigenvalues of the ${\alpha n \times \alpha n}$ minor to be equal to ${-1}$ (resp. ${+1}$). For ${\alpha = e^{-t} \leq 1/2}$, the poles disappear and one just has

$\displaystyle \mu_\alpha = \frac{1}{2\pi \alpha(1-x^2)} (4\alpha(1-\alpha)-x^2)_+^{1/2}\ dx.$

For ${\alpha=1/2}$, one has an inverse semicircle distribution

$\displaystyle \mu_{1/2} = \frac{1}{\pi} (1-x^2)_+^{-1/2}.$

There is presumably a direct geometric explanation of this fact (basically describing the singular values of the product of two random orthogonal projections to half-dimensional subspaces of ${{\bf C}^n}$), but I do not know of one off-hand.

The evolution of ${s(t,z)}$ can also be understood using the ${R}$-transform and ${S}$-transform from free probability. Formally, letlet ${z(t,s)}$ be the inverse of ${s(t,z)}$, thus

$\displaystyle s(t,z(t,s)) = s$

for all ${t,s}$, and then define the ${R}$-transform

$\displaystyle R(t,s) := z(t,-s) - \frac{1}{s}.$

The equation (9) may be rewritten as

$\displaystyle z( t, e^t s ) = z(0,s) + s^{-1} (1-e^{-t})$

and hence

$\displaystyle R(t, -e^t s) = R(0, -s)$

or equivalently

$\displaystyle R(t,s) = R(0, e^{-t} s). \ \ \ \ \ (12)$

See these previous notes for a discussion of free probability topics such as the ${R}$-transform.

Example 6 If ${s(t,z) = \frac{1}{c-z}}$ then the ${R}$ transform is ${R(t,s) = c}$.

Example 7 If ${s(t,z)}$ is given by (10), then the ${R}$ transform is

$\displaystyle R(t,s) = \sigma^2 e^{-t} s.$

Example 8 If ${s(t,z)}$ is given by (11), then the ${R}$ transform is

$\displaystyle R(t,s) = \frac{-1 + \sqrt{1 + 4 s^2 e^{-2t}}}{2 s e^{-t}}.$

This simple relationship (12) is essentially due to Nica and Speicher (thanks to Dima Shylakhtenko for this reference). It has the remarkable consequence that when ${\alpha = 1/m}$ is the reciprocal of a natural number ${m}$, then ${\mu_{1/m}}$ is the free arithmetic mean of ${m}$ copies of ${\mu}$, that is to say ${\mu_{1/m}}$ is the free convolution ${\mu \boxplus \dots \boxplus \mu}$ of ${m}$ copies of ${\mu}$, pushed forward by the map ${\lambda \rightarrow \lambda/m}$. In terms of random matrices, this is asserting that the top ${n/m \times n/m}$ minor of a random matrix ${M}$ has spectral measure approximately equal to that of an arithmetic mean ${\frac{1}{m} (M_1 + \dots + M_m)}$ of ${m}$ independent copies of ${M}$, so that the process of taking top left minors is in some sense a continuous analogue of the process of taking freely independent arithmetic means. There ought to be a geometric proof of this assertion, but I do not know of one. In the limit ${m \rightarrow \infty}$ (or ${\alpha \rightarrow 0}$), the ${R}$-transform becomes linear and the spectral measure becomes semicircular, which is of course consistent with the free central limit theorem.

In a similar vein, if one defines the function

$\displaystyle \omega(t,z) := \alpha \int_{\bf R} \frac{zx}{1-zx}\ d\mu_\alpha(x) = e^{-t} (- 1 - z^{-1} s(t, z^{-1}))$

and inverts it to obtain a function ${z(t,\omega)}$ with

$\displaystyle \omega(t, z(t,\omega)) = \omega$

for all ${t, \omega}$, then the ${S}$-transform ${S(t,\omega)}$ is defined by

$\displaystyle S(t,\omega) := \frac{1+\omega}{\omega} z(t,\omega).$

Writing

$\displaystyle s(t,z) = - z^{-1} ( 1 + e^t \omega(t, z^{-1}) )$

for any ${t}$, ${z}$, we have

$\displaystyle z_0 + s(0,z_0)^{-1} (1-e^{-t}) = z_0 \frac{\omega(0,z_0^{-1})+e^{-t}}{\omega(0,z_0^{-1})+1}$

and so (9) becomes

$\displaystyle - z_0^{-1} \frac{\omega(0,z_0^{-1})+1}{\omega(0,z_0^{-1})+e^{-t}} (1 + e^{t} \omega(t, z_0^{-1} \frac{\omega(0,z_0^{-1})+1}{\omega(0,z_0^{-1})+e^{-t}}))$

$\displaystyle = - e^t z_0^{-1} (1 + \omega(0, z_0^{-1}))$

which simplifies to

$\displaystyle \omega(t, z_0^{-1} \frac{\omega(0,z_0^{-1})+1}{\omega(0,z_0^{-1})+e^{-t}})) = \omega(0, z_0^{-1});$

replacing ${z_0}$ by ${z(0,\omega)^{-1}}$ we obtain

$\displaystyle \omega(t, z(0,\omega) \frac{\omega+1}{\omega+e^{-t}}) = \omega$

and thus

$\displaystyle z(0,\omega)\frac{\omega+1}{\omega+e^{-t}} = z(t, \omega)$

and hence

$\displaystyle S(0, \omega) = \frac{\omega+e^{-t}}{\omega+1} S(t, \omega).$

One can compute ${\frac{\omega+e^{-t}}{\omega+1}}$ to be the ${S}$-transform of the measure ${(1-\alpha) \delta_0 + \alpha \delta_1}$; from the link between ${S}$-transforms and free products (see e.g. these notes of Guionnet), we conclude that ${(1-\alpha)\delta_0 + \alpha \mu_\alpha}$ is the free product of ${\mu_1}$ and ${(1-\alpha) \delta_0 + \alpha \delta_1}$. This is consistent with the random matrix theory interpretation, since ${(1-\alpha)\delta_0 + \alpha \mu_\alpha}$ is also the spectral measure of ${PMP}$, where ${P}$ is the orthogonal projection to the span of the first ${\alpha n}$ basis elements, so in particular ${P}$ has spectral measure ${(1-\alpha) \delta_0 + \alpha \delta_1}$. If ${M}$ is unitarily invariant then (by a fundamental result of Voiculescu) it is asymptotically freely independent of ${P}$, so the spectral measure of ${PMP = P^{1/2} M P^{1/2}}$ is asymptotically the free product of that of ${M}$ and of ${P}$.

Fix a non-negative integer ${k}$. Define an (weak) integer partition of length ${k}$ to be a tuple ${\lambda = (\lambda_1,\dots,\lambda_k)}$ of non-increasing non-negative integers ${\lambda_1 \geq \dots \geq \lambda_k \geq 0}$. (Here our partitions are “weak” in the sense that we allow some parts of the partition to be zero. Henceforth we will omit the modifier “weak”, as we will not need to consider the more usual notion of “strong” partitions.) To each such partition ${\lambda}$, one can associate a Young diagram consisting of ${k}$ left-justified rows of boxes, with the ${i^{th}}$ row containing ${\lambda_i}$ boxes. A semi-standard Young tableau (or Young tableau for short) ${T}$ of shape ${\lambda}$ is a filling of these boxes by integers in ${\{1,\dots,k\}}$ that is weakly increasing along rows (moving rightwards) and strictly increasing along columns (moving downwards). The collection of such tableaux will be denoted ${{\mathcal T}_\lambda}$. The weight ${|T|}$ of a tableau ${T}$ is the tuple ${(n_1,\dots,n_k)}$, where ${n_i}$ is the number of occurrences of the integer ${i}$ in the tableau. For instance, if ${k=3}$ and ${\lambda = (6,4,2)}$, an example of a Young tableau of shape ${\lambda}$ would be

$\displaystyle \begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 2 & 3 & 3 \\ \cline{1-6} 2 & 2 & 2 &3\\ \cline{1-4} 3 & 3\\ \cline{1-2} \end{tabular}$

The weight here would be ${|T| = (3,4,5)}$.

To each partition ${\lambda}$ one can associate the Schur polynomial ${s_\lambda(u_1,\dots,u_k)}$ on ${k}$ variables ${u = (u_1,\dots,u_k)}$, which we will define as

$\displaystyle s_\lambda(u) := \sum_{T \in {\mathcal T}_\lambda} u^{|T|}$

using the multinomial convention

$\displaystyle (u_1,\dots,u_k)^{(n_1,\dots,n_k)} := u_1^{n_1} \dots u_k^{n_k}.$

Thus for instance the Young tableau ${T}$ given above would contribute a term ${u_1^3 u_2^4 u_3^5}$ to the Schur polynomial ${s_{(6,4,2)}(u_1,u_2,u_3)}$. In the case of partitions of the form ${(n,0,\dots,0)}$, the Schur polynomial ${s_{(n,0,\dots,0)}}$ is just the complete homogeneous symmetric polynomial ${h_n}$ of degree ${n}$ on ${k}$ variables:

$\displaystyle s_{(n,0,\dots,0)}(u_1,\dots,u_k) := \sum_{n_1,\dots,n_k \geq 0: n_1+\dots+n_k = n} u_1^{n_1} \dots u_k^{n_k},$

thus for instance

$\displaystyle s_{(3,0)}(u_1,u_2) = u_1^3 + u_1^2 u_2 + u_1 u_2^2 + u_2^3.$

Schur polyomials are ubiquitous in the algebraic combinatorics of “type ${A}$ objects” such as the symmetric group ${S_k}$, the general linear group ${GL_k}$, or the unitary group ${U_k}$. For instance, one can view ${s_\lambda}$ as the character of an irreducible polynomial representation of ${GL_k({\bf C})}$ associated with the partition ${\lambda}$. However, we will not focus on these interpretations of Schur polynomials in this post.

This definition of Schur polynomials allows for a way to describe the polynomials recursively. If ${k > 1}$ and ${T}$ is a Young tableau of shape ${\lambda = (\lambda_1,\dots,\lambda_k)}$, taking values in ${\{1,\dots,k\}}$, one can form a sub-tableau ${T'}$ of some shape ${\lambda' = (\lambda'_1,\dots,\lambda'_{k-1})}$ by removing all the appearances of ${k}$ (which, among other things, necessarily deletes the ${k^{th}}$ row). For instance, with ${T}$ as in the previous example, the sub-tableau ${T'}$ would be

$\displaystyle \begin{tabular}{|c|c|c|c|} \hline 1 & 1 & 1 & 2 \\ \cline{1-4} 2 & 2 & 2 \\ \cline{1-3} \end{tabular}$

and the reduced partition ${\lambda'}$ in this case is ${(4,3)}$. As Young tableaux are required to be strictly increasing down columns, we can see that the reduced partition ${\lambda'}$ must intersperse the original partition ${\lambda}$ in the sense that

$\displaystyle \lambda_{i+1} \leq \lambda'_i \leq \lambda_i \ \ \ \ \ (1)$

for all ${1 \leq i \leq k-1}$; we denote this interspersion relation as ${\lambda' \prec \lambda}$ (though we caution that this is not intended to be a partial ordering). In the converse direction, if ${\lambda' \prec \lambda}$ and ${T'}$ is a Young tableau with shape ${\lambda'}$ with entries in ${\{1,\dots,k-1\}}$, one can form a Young tableau ${T}$ with shape ${\lambda}$ and entries in ${\{1,\dots,k\}}$ by appending to ${T'}$ an entry of ${k}$ in all the boxes that appear in the ${\lambda}$ shape but not the ${\lambda'}$ shape. This one-to-one correspondence leads to the recursion

$\displaystyle s_\lambda(u) = \sum_{\lambda' \prec \lambda} s_{\lambda'}(u') u_k^{|\lambda| - |\lambda'|} \ \ \ \ \ (2)$

where ${u = (u_1,\dots,u_k)}$, ${u' = (u_1,\dots,u_{k-1})}$, and the size ${|\lambda|}$ of a partition ${\lambda = (\lambda_1,\dots,\lambda_k)}$ is defined as ${|\lambda| := \lambda_1 + \dots + \lambda_k}$.

One can use this recursion (2) to prove some further standard identities for Schur polynomials, such as the determinant identity

$\displaystyle s_\lambda(u) V(u) = \det( u_i^{\lambda_j+k-j} )_{1 \leq i,j \leq k} \ \ \ \ \ (3)$

for ${u=(u_1,\dots,u_k)}$, where ${V(u)}$ denotes the Vandermonde determinant

$\displaystyle V(u) := \prod_{1 \leq i < j \leq k} (u_i - u_j), \ \ \ \ \ (4)$

or the Jacobi-Trudi identity

$\displaystyle s_\lambda(u) = \det( h_{\lambda_j - j + i}(u) )_{1 \leq i,j \leq k}, \ \ \ \ \ (5)$

with the convention that ${h_d(u) = 0}$ if ${d}$ is negative. Thus for instance

$\displaystyle s_{(1,1,0,\dots,0)}(u) = h_1^2(u) - h_0(u) h_2(u) = \sum_{1 \leq i < j \leq k} u_i u_j.$

We review the (standard) derivation of these identities via (2) below the fold. Among other things, these identities show that the Schur polynomials are symmetric, which is not immediately obvious from their definition.

One can also iterate (2) to write

$\displaystyle s_\lambda(u) = \sum_{() = \lambda^0 \prec \lambda^1 \prec \dots \prec \lambda^k = \lambda} \prod_{j=1}^k u_j^{|\lambda^j| - |\lambda^{j-1}|} \ \ \ \ \ (6)$

where the sum is over all tuples ${\lambda^1,\dots,\lambda^k}$, where each ${\lambda^j}$ is a partition of length ${j}$ that intersperses the next partition ${\lambda^{j+1}}$, with ${\lambda^k}$ set equal to ${\lambda}$. We will call such a tuple an integral Gelfand-Tsetlin pattern based at ${\lambda}$.

One can generalise (6) by introducing the skew Schur functions

$\displaystyle s_{\lambda/\mu}(u) := \sum_{\mu = \lambda^i \prec \dots \prec \lambda^k = \lambda} \prod_{j=i+1}^k u_j^{|\lambda^j| - |\lambda^{j-1}|} \ \ \ \ \ (7)$

for ${u = (u_{i+1},\dots,u_k)}$, whenever ${\lambda}$ is a partition of length ${k}$ and ${\mu}$ a partition of length ${i}$ for some ${0 \leq i \leq k}$, thus the Schur polynomial ${s_\lambda}$ is also the skew Schur polynomial ${s_{\lambda /()}}$ with ${i=0}$. (One could relabel the variables here to be something like ${(u_1,\dots,u_{k-i})}$ instead, but this labeling seems slightly more natural, particularly in view of identities such as (8) below.)

By construction, we have the decomposition

$\displaystyle s_{\lambda/\nu}(u_{i+1},\dots,u_k) = \sum_\mu s_{\mu/\nu}(u_{i+1},\dots,u_j) s_{\lambda/\mu}(u_{j+1},\dots,u_k) \ \ \ \ \ (8)$

whenever ${0 \leq i \leq j \leq k}$, and ${\nu, \mu, \lambda}$ are partitions of lengths ${i,j,k}$ respectively. This gives another recursive way to understand Schur polynomials and skew Schur polynomials. For instance, one can use it to establish the generalised Jacobi-Trudi identity

$\displaystyle s_{\lambda/\mu}(u) = \det( h_{\lambda_j - j - \mu_i + i}(u) )_{1 \leq i,j \leq k}, \ \ \ \ \ (9)$

with the convention that ${\mu_i = 0}$ for ${i}$ larger than the length of ${\mu}$; we do this below the fold.

The Schur polynomials (and skew Schur polynomials) are “discretised” (or “quantised”) in the sense that their parameters ${\lambda, \mu}$ are required to be integer-valued, and their definition similarly involves summation over a discrete set. It turns out that there are “continuous” (or “classical”) analogues of these functions, in which the parameters ${\lambda,\mu}$ now take real values rather than integers, and are defined via integration rather than summation. One can view these continuous analogues as a “semiclassical limit” of their discrete counterparts, in a manner that can be made precise using the machinery of geometric quantisation, but we will not do so here.

The continuous analogues can be defined as follows. Define a real partition of length ${k}$ to be a tuple ${\lambda = (\lambda_1,\dots,\lambda_k)}$ where ${\lambda_1 \geq \dots \geq \lambda_k \geq 0}$ are now real numbers. We can define the relation ${\lambda' \prec \lambda}$ of interspersion between a length ${k-1}$ real partition ${\lambda' = (\lambda'_1,\dots,\lambda'_{k-1})}$ and a length ${k}$ real partition ${\lambda = (\lambda_1,\dots,\lambda_{k})}$ precisely as before, by requiring that the inequalities (1) hold for all ${1 \leq i \leq k-1}$. We can then define the continuous Schur functions ${S_\lambda(x)}$ for ${x = (x_1,\dots,x_k) \in {\bf R}^k}$ recursively by defining

$\displaystyle S_{()}() = 1$

and

$\displaystyle S_\lambda(x) = \int_{\lambda' \prec \lambda} S_{\lambda'}(x') \exp( (|\lambda| - |\lambda'|) x_k ) \ \ \ \ \ (10)$

for ${k \geq 1}$ and ${\lambda}$ of length ${k}$, where ${x' := (x_1,\dots,x_{k-1})}$ and the integral is with respect to ${k-1}$-dimensional Lebesgue measure, and ${|\lambda| = \lambda_1 + \dots + \lambda_k}$ as before. Thus for instance

$\displaystyle S_{(\lambda_1)}(x_1) = \exp( \lambda_1 x_1 )$

and

$\displaystyle S_{(\lambda_1,\lambda_2)}(x_1,x_2) = \int_{\lambda_2}^{\lambda_1} \exp( \lambda'_1 x_1 + (\lambda_1+\lambda_2-\lambda'_1) x_2 )\ d\lambda'_1.$

More generally, we can define the continuous skew Schur functions ${S_{\lambda/\mu}(x)}$ for ${\lambda}$ of length ${k}$, ${\mu}$ of length ${j \leq k}$, and ${x = (x_{j+1},\dots,x_k) \in {\bf R}^{k-j}}$ recursively by defining

$\displaystyle S_{\mu/\mu}() = 1$

and

$\displaystyle S_{\lambda/\mu}(x) = \int_{\lambda' \prec \lambda} S_{\lambda'/\mu}(x') \exp( (|\lambda| - |\lambda'|) x_k )$

for ${k > j}$. Thus for instance

$\displaystyle S_{(\lambda_1,\lambda_2,\lambda_3)/(\mu_1,\mu_2)}(x_3) = 1_{\lambda_3 \leq \mu_2 \leq \lambda_2 \leq \mu_1 \leq \lambda_1} \exp( x_3 (\lambda_1+\lambda_2+\lambda_3 - \mu_1 - \mu_2 ))$

and

$\displaystyle S_{(\lambda_1,\lambda_2,\lambda_3)/(\mu_1)}(x_2, x_3) = \int_{\lambda_2 \leq \lambda'_2 \leq \lambda_2, \mu_1} \int_{\mu_1, \lambda_2 \leq \lambda'_1 \leq \lambda_1}$

$\displaystyle \exp( x_2 (\lambda'_1+\lambda'_2 - \mu_1) + x_3 (\lambda_1+\lambda_2+\lambda_3 - \lambda'_1 - \lambda'_2))\ d\lambda'_1 d\lambda'_2.$

By expanding out the recursion, one obtains the analogue

$\displaystyle S_\lambda(x) = \int_{\lambda^1 \prec \dots \prec \lambda^k = \lambda} \exp( \sum_{j=1}^k x_j (|\lambda^j| - |\lambda^{j-1}|))\ d\lambda^1 \dots d\lambda^{k-1},$

of (6), and more generally one has

$\displaystyle S_{\lambda/\mu}(x) = \int_{\mu = \lambda^i \prec \dots \prec \lambda^k = \lambda} \exp( \sum_{j=i+1}^k x_j (|\lambda^j| - |\lambda^{j-1}|))\ d\lambda^{i+1} \dots d\lambda^{k-1}.$

We will call the tuples ${(\lambda^1,\dots,\lambda^k)}$ in the first integral real Gelfand-Tsetlin patterns based at ${\lambda}$. The analogue of (8) is then

$\displaystyle S_{\lambda/\nu}(x_{i+1},\dots,x_k) = \int S_{\mu/\nu}(x_{i+1},\dots,x_j) S_{\lambda/\mu}(x_{j+1},\dots,x_k)\ d\mu$

where the integral is over all real partitions ${\mu}$ of length ${j}$, with Lebesgue measure.

By approximating various integrals by their Riemann sums, one can relate the continuous Schur functions to their discrete counterparts by the limiting formula

$\displaystyle N^{-k(k-1)/2} s_{\lfloor N \lambda \rfloor}( \exp[ x/N ] ) \rightarrow S_\lambda(x) \ \ \ \ \ (11)$

as ${N \rightarrow \infty}$ for any length ${k}$ real partition ${\lambda = (\lambda_1,\dots,\lambda_k)}$ and any ${x = (x_1,\dots,x_k) \in {\bf R}^k}$, where

$\displaystyle \lfloor N \lambda \rfloor := ( \lfloor N \lambda_1 \rfloor, \dots, \lfloor N \lambda_k \rfloor )$

and

$\displaystyle \exp[x/N] := (\exp(x_1/N), \dots, \exp(x_k/N)).$

More generally, one has

$\displaystyle N^{j(j-1)/2-k(k-1)/2} s_{\lfloor N \lambda \rfloor / \lfloor N \mu \rfloor}( \exp[ x/N ] ) \rightarrow S_{\lambda/\mu}(x)$

as ${N \rightarrow \infty}$ for any length ${k}$ real partition ${\lambda}$, any length ${j}$ real partition ${\mu}$ with ${0 \leq j \leq k}$, and any ${x = (x_{j+1},\dots,x_k) \in {\bf R}^{k-j}}$.

As a consequence of these limiting formulae, one expects all of the discrete identities above to have continuous counterparts. This is indeed the case; below the fold we shall prove the discrete and continuous identities in parallel. These are not new results by any means, but I was not able to locate a good place in the literature where they are explicitly written down, so I thought I would try to do so here (primarily for my own internal reference, but perhaps the calculations will be worthwhile to some others also).

Israel Gelfand, who made profound and prolific contributions to many areas of mathematics, including functional analysis, representation theory, operator algebras, and partial differential equations, died on Monday, age 96.

Gelfand’s beautiful theory of ${C^*}$-algebras and related spaces made quite an impact on me as a graduate student in Princeton, to the point where I was seriously considering working in this area; but there was not much activity in operator algebras at the time there, and I ended up working in harmonic analysis under Eli Stein instead. (Though I am currently involved in another operator algebras project, of which I hope to be able to discuss in the near future. The commutative version of Gelfand’s theory is discussed in these lecture notes of mine.)

I met Gelfand only once, in one of the famous “Gelfand seminars” at the IHES in 2000. The speaker was Tim Gowers, on his new proof of Szemerédi’s theorem. (Endre Szemerédi, incidentally, was Gelfand’s student.) Gelfand’s introduction to the seminar, on the subject of Banach spaces which both mathematicians contributed so greatly to, was approximately as long as Gowers’ talk itself!

There are far too many contributions to mathematics by Gelfand to name here, so I will only mention two. The first are the Gelfand-Tsetlin patterns, induced by an ${n \times n}$ Hermitian matrix ${A}$. Such matrices have ${n}$ real eigenvalues ${\lambda_{n,1} \leq \ldots \leq \lambda_{n,n}}$. If one takes the top ${n-1 \times n-1}$ minor, this is another Hermitian matrix, whose ${n-1}$ eigenvalues ${\lambda_{n-1,1} \leq \ldots \leq \lambda_{n-1,n-1}}$ intersperse the ${n}$ eigenvalues of the original matrix: ${\lambda_{n,i} \leq \lambda_{n-1,i} \leq \lambda_{n,i+1}}$ for every ${1 \leq i \leq n-1}$. This interspersing can be easily seen from the minimax characterisation

$\displaystyle \lambda_{n,i} = \inf_{\hbox{dim}(V)=i} \sup_{v \in V: \|v\|=1} \langle Av, v \rangle$

of the eigenvalues of ${A}$, with the eigenvalues of the ${n-1 \times n-1}$ minor being similar but with ${V}$ now restricted to a subspace of ${{\mathbb C}^{n-1}}$ rather than ${{\mathbb C}^n}$.

Similarly, the eigenvalues ${\lambda_{n-2,1} \leq \ldots \leq \lambda_{n-2,n-2}}$ of the top ${n-2 \times n-2}$ minor of ${A}$ intersperse those of the previous minor. Repeating this procedure one eventually gets a pyramid of real numbers of height and width ${n}$, with the numbers in each row interspersing the ones in the row below. Such a pattern is known as a Gelfand-Tsetlin pattern. The space of such patterns forms a convex cone, and (if one fixes the initial eigenvalues ${\lambda_{n,1},\ldots,\lambda_{n,n}}$) becomes a compact convex polytope. If one fixes the initial eigenvalues ${\lambda_{n,1},\ldots,\lambda_{n,n}}$ of ${A}$ but chooses the eigenvectors randomly (using the Haar measure of the unitary group), then the resulting Gelfand-Tsetlin pattern is uniformly distributed across this polytope; the ${n=2}$ case of this observation is essentially the classic observation of Archimedes that the cross-sectional areas of a sphere and a circumscribing cylinder are the same. (Ultimately, the reason for this is that the Gelfand-Tsetlin pattern almost turns the space of all ${A}$ with a fixed spectrum (i.e. the co-adjoint orbit associated to that spectrum) into a toric variety. More precisely, there exists a mostly diffeomorphic map from the co-adjoint orbit to a (singular) toric variety, and the Gelfand-Tsetlin pattern induces a complete set of action variables on that variety.) There is also a “quantum” (or more precisely, representation-theoretic) version of this observation, in which one can decompose any irreducible representation of the unitary group ${U(n)}$ into a canonical basis (the Gelfand-Tsetlin basis), indexed by integer-valued Gelfand-Tsetlin patterns, by first decomposing this representation into irreducible representations of ${U(n-1)}$, then ${U(n-2)}$, and so forth.

The structure, symplectic geometry, and representation theory of Gelfand-Tsetlin patterns was enormously influential in my own work with Allen Knutson on honeycomb patterns, which control the sums of Hermitian matrices and also the structure constants of the tensor product operation for representations of ${U(n)}$; indeed, Gelfand-Tsetlin patterns arise as the degenerate limit of honeycombs in three different ways, and we in fact discovered honeycombs by trying to glue three Gelfand-Tsetlin patterns together. (See for instance our Notices article for more discussion. The honeycomb analogue of the representation-theoretic properties of these patterns was eventually established by Henriques and Kamnitzer, using ${gl(n)}$ crystals and their Kashiwara bases.)

The second contribution of Gelfand I want to discuss is the Gelfand-Levitan-Marchenko equation for solving the one-dimensional inverse scattering problem: given the scattering data of an unknown potential function ${V(x)}$, recover ${V}$. This is already interesting in and of itself, but is also instrumental in solving integrable systems such as the Korteweg-de Vries equation, because the Lax pair formulation of such equations implies that they can be linearised (and solved explicitly) by applying the scattering and inverse scattering transforms associated with the Lax operator. I discuss the derivation of this equation below the fold.