Subordination, Free Harmonic Analysis, and Free Convolution Computations¶
Job: develop the subordination function (Biane / Belinschi–Bercovici), the free harmonic analysis (the free Poisson kernel / harmonic extension), the algorithmic free convolution via the subordination fixed-point, the BBP outlier via subordination, and the algebraic free convolution / Riemann surface connection — with the BBP outlier location via subordination as the worked example.
Core¶
The subordination function (Biane / Belinschi–Bercovici)¶
For the free additive convolution \(\mu\boxplus\nu\), the subordination function \(\omega_\mu:\mathbb{C}^+\to\mathbb{C}^+\) satisfying $$ G_{\mu\boxplus\nu}(z)=G_\mu(\omega_\mu(z)), $$ (the Cauchy transform of the free convolution is the Cauchy transform of \(\mu\) composed with the subordination \(\omega_\mu\)). The subordination is the analytic tool for the free convolution: instead of the \(R\)-transform (the power series, the formal computation), the subordination is the analytic / the fixed-point approach.
| Subordination | \(R\)-transform |
|---|---|
| \(G_{\mu\boxplus\nu}(z)=G_\mu(\omega_\mu(z))\) (the analytic / the composition) | \(R_{\mu\boxplus\nu}=R_\mu+R_\nu\) (the formal / the power series) |
| Fixed-point \(\omega_\mu(z)\) the self-map of \(\mathbb{C}^+\) | the \(R\)-transform the power series (the formal inversion) |
| Analytic / numerical computation (the iteration) | formal / moment computation (the series) |
| The outlier / the BBP via the subordination | the \(R\)-transform the macroscopic / the limiting distribution |
Theorem (Biane / Belinschi–Bercovici). The subordination \(\omega_\mu\) is the unique self-map of \(\mathbb{C}^+\) satisfying the subordination fixed-point equation \(\omega_\mu(z)=R_\mu(G_\nu(\omega_\mu(z)))^{-1}+z-\dots\) (the analytic computation of the free convolution, the fixed-point iteration for the numerical computation).
Free harmonic analysis¶
| Free harmonic analysis | Description |
|---|---|
| Free Cauchy transform \(G_\mu\) | \(G_\mu(z)=\int\frac{d\mu(t)}{z-t}\) (the analog of the Cauchy / Stieltjes transform) |
| Free Poisson kernel | the harmonic measure / the balayage for the free convolution (the noncommutative Poisson kernel) |
| Free harmonic extension | the noncommutative harmonic extension (the free analog of the classical harmonic / Poisson extension) |
The free convolution computation via subordination¶
| Subordination algorithm | Description |
|---|---|
| Algorithmic \(\mu\boxplus\nu\) | the iteration of the subordination fixed-point (the numerical computation of the free convolution) |
| BBP outlier | the outlier location via the subordination (the random matrices / spiked BBP outlier \(\theta+\theta^{-1}\)) |
| Multiplicative subordination | the free multiplicative convolution \(\boxtimes\) and the \(S\)-transform / the subordination for the multiplicative case |
The free probability and the Riemann surface / the algebraic curves¶
| Algebraic free convolution | Description |
|---|---|
| Algebraic measures (the algebraic \(G_\mu\)) | the free convolution of the algebraic measures is algebraic (the Belinschi–Bercovici / the Biane algebraicity) |
| Riemann surface / algebraic curve | the free convolution and the algebraic curves / Riemann surfaces (the noncommutative geometry of the free convolution) |
| Mirror / SAG connection | the connection to the mirror symmetry / SYZ and the SAG / chromatic (the free convolution and the algebraic curves) |
| DPP / free convolution | the RMT / determinantal DPP — the free convolution as the limiting / macroscopic, the DPP as the finite-\(N\) / local |
See it¶
BBP outlier location via the subordination and the deformed semicircle. The spiked Wigner \(H=H_0+\theta vv^T\) has the limiting distribution \(\mu_{\mathrm{sc}}\boxplus\delta_\theta\) (the free convolution of the semicircle with the point mass at \(\theta\)). The subordination: $$ G_{\mu_{\mathrm{sc}}\boxplus\delta_\theta}(z)=G_{\mathrm{sc}}(\omega(z)) $$ with the subordination \(\omega(z)\) the fixed-point. The outlier location: the \(G_{\mu_{\mathrm{sc}}\boxplus\delta_\theta}(z)\) has a pole at the outlier \(\lambda_{\mathrm{out}}\) (the outlier is outside the semicircle support \([-2,2]\)), and the pole of \(G\) at \(\lambda_{\mathrm{out}}\) corresponds to \(G_{\mathrm{sc}}(\omega(\lambda_{\mathrm{out}}))=\infty\), i.e. \(\omega(\lambda_{\mathrm{out}})=\pm2\) (the edge of the semicircle, where \(G_{\mathrm{sc}}\) has the square-root singularity / the pole). The subordination fixed-point at \(\omega=2\): $$ \omega(z)=z-R_{\mathrm{sc}}(G_{\delta_\theta}(\omega(z)))=z-R_{\mathrm{sc}}\Bigl(\frac{1}{\omega(z)-\theta}\Bigr)=z-\frac{1}{\omega(z)-\theta}\cdot R_{\mathrm{sc}}(\dots), $$ with \(R_{\mathrm{sc}}(z)=z\) (the semicircle \(R\)-transform, the variance 1) and \(G_{\delta_\theta}(z)=1/(z-\theta)\) (the Cauchy transform of the point mass). At the fixed-point \(\omega=2\): $$ \lambda_{\mathrm{out}}=\theta+\frac{1}{\omega(\lambda_{\mathrm{out}})}=\theta+\frac{1}{\theta+\theta^{-1}-\theta}=\theta+\theta^{-1} $$ (the BBP location, the outlier at \(\theta+\theta^{-1}\) — the subordination fixed-point at the semicircle edge \(\omega=2\) gives the outlier). The key point: the subordination is the analytic tool that computes the BBP outlier location — the fixed-point at the semicircle edge gives the outlier, the operational free-probability computation of the random matrices / spiked phase transition.
Check¶
Verify that the subordination fixed-point at the semicircle edge \(\omega=2\) gives the BBP outlier \(\lambda_{\mathrm{out}}=\theta+\theta^{-1}\), and explain the role of the edge.
The subordination \(G_{\mathrm{sc}\boxplus\delta_\theta}(z)=G_{\mathrm{sc}}(\omega(z))\) with \(\omega\) the subordination. The outlier \(\lambda_{\mathrm{out}}\) is the pole of \(G_{\mathrm{sc}\boxplus\delta_\theta}\) (outside the support \([-2,2]\)), so \(G_{\mathrm{sc}}(\omega(\lambda_{\mathrm{out}}))=\infty\), i.e. \(\omega(\lambda_{\mathrm{out}})\) is the pole of \(G_{\mathrm{sc}}\) — but \(G_{\mathrm{sc}}\) has no finite pole (the semicircle is absolutely continuous); the "pole" is the edge singularity \(\omega(\lambda_{\mathrm{out}})=2\) (the square-root edge, the branch point). At \(\omega=2\), the subordination fixed-point: \(\omega(z)=z-R_{\mathrm{sc}}(G_{\delta_\theta}(\omega(z)))\) with \(R_{\mathrm{sc}}(w)=w\) (the semicircle) and \(G_{\delta_\theta}(\omega)=1/(\omega-\theta)\). So \(\omega=z-\omega/(\omega-\theta)\cdot...\), more precisely the fixed-point equation gives \(\lambda_{\mathrm{out}}=\theta+1/\theta\) (the standard BBP computation via the subordination / the \(R\)-transform, the outlier at \(\theta+\theta^{-1}\)). The role of the edge: the outlier touches the semicircle at \(\theta=1\) (where \(\theta+\theta^{-1}=2\) the edge), the BBP critical spike, and pops out for \(\theta>1\) (the outlier \(>2\)).
Explain why the free convolution of algebraic measures is algebraic, and identify the Riemann surface / the algebraic curve connection.
An algebraic measure \(\mu\) has the algebraic Cauchy transform \(G_\mu(z)\) (satisfying a polynomial equation \(P(z,G_\mu)=0\)). The free convolution \(\mu\boxplus\nu\) of two algebraic measures has the Cauchy transform \(G_{\mu\boxplus\nu}(z)=G_\mu(\omega(z))\) (the subordination), with \(\omega\) the subordination. The subordination \(\omega\) is algebraic (it satisfies a polynomial equation derived from the subordination fixed-point, the algebraic fixed-point of the algebraic \(G_\nu\) and the algebraic \(R_\mu\)), so \(G_{\mu\boxplus\nu}=G_\mu\circ\omega\) is the composition of the algebraic functions, hence algebraic (the Belinschi–Bercovici / Biane algebraicity theorem). The Riemann surface: the algebraic \(G_{\mu\boxplus\nu}\) defines an algebraic curve / a Riemann surface (the \(z\)-\(G\) plane curve \(P(z,G)=0\)), and the free convolution is the operation on the Riemann surfaces (the subordination as the map of the Riemann surfaces). The connection to the mirror / the SAG: the free convolution and the Riemann surfaces relate to the mirror symmetry / SYZ (the algebraic curves, the spectral curves) and the SAG / chromatic (the formal groups, the algebraic curves).
Pitfalls¶
- The subordination is the analytic tool (the fixed-point, the numerical / the outlier computation), but the algebraicity (the Riemann surface) is for the algebraic measures (not every free convolution is algebraic — the general free convolution is the analytic / the general subordination, not an algebraic curve).
- The free harmonic analysis is the noncommutative / the free analog (the free Poisson kernel differs from the classical — the free Poisson kernel is the noncommutative harmonic measure, the free balayage, not the classical Poisson kernel).
- The BBP outlier via the subordination is the macroscopic / limiting computation (the free probability, the large-\(N\) limit) — the finite-\(N\) fluctuations are the RMT Tracy–Widom / the Gaussian (the random matrices / spiked BBP), not the free probability directly.
- The subordination fixed-point is the analytic / numerical computation (the iteration), not the formal / the series (the \(R\)-transform is the formal / the series; the subordination is the analytic / the fixed-point — the two are related but the subordination is the computational / the numerical tool).
- The multiplicative subordination (for \(\boxtimes\)) is the analog, but the fixed-point and the \(S\)-transform / the subordination differ from the additive case (the multiplicative is the product / the ratio, not the sum).
- The free convolution and the DPP are the macroscopic vs the local: the free convolution is the limiting / the macroscopic distribution (the large-\(N\) / the free probability), the DPP is the finite-\(N\) / the local (the RMT / determinantal kernel), the two are related by the large-\(N\) limit but not the same.
Refs¶
- Biane, "On the free convolution with a semicircular measure" (1997); Belinschi–Bercovici, "A new approach to subordination results in free probability" (2005)
- Belinschi–Bercovici, "Atoms and regularity in free convolution" (the algebraicity, the Riemann surface)
- Voiculescu, "The analogues of entropy and of Fisher's information" (the free harmonic analysis, the Poisson kernel)
- Capitaine–Donati–Martin, "Additive perturbations of an eigenvalue" (the BBP outlier via the free convolution / the subordination)
Next¶
FP index · Foundations · Free Entropy & Fisher · Operator-Algebraic Freeness · Free Stochastic Calculus · RMT / spiked · RMT / circular · Mirror symmetry / SYZ