Determinantal Point Processes and β-Ensembles¶
Job: develop determinantal point processes (DPP), the GUE kernel / orthogonal polynomial / Riemann–Hilbert method, the Airy kernel / Tracy–Widom / Painlevé II, and the \(\beta\)-ensembles / loop equations / topological expansion / stochastic Airy operator — with the GUE kernel → Airy → Tracy–Widom computation as the worked example.
Core¶
Determinantal point processes (DPP)¶
A point process \(\{x_i\}\) on \(\Lambda\) is determinantal with kernel \(K\) if the \(k\)-point correlation functions are $$ \rho_k(x_1,\dots,x_k)=\det(K(x_i,x_j))_{i,j=1}^k. $$ The GUE / CUE eigenvalue processes are DPP. The kernel \(K\) is a projection / a trace-class integral operator on \(L^2(\Lambda)\), with \(\#\{x_i\}=\mathrm{Tr}\,K\) (the number of points = the trace / the rank).
| DPP data | Description |
|---|---|
| Kernel \(K(x,y)\) | the correlation functions \(\rho_k=\det(K(x_i,x_j))\) |
| Projection / trace-class | \(K\) is a projection (\(K^2=K\)) or trace-class on \(L^2(\Lambda)\) |
| Number of points | \(\#\{x_i\}=\mathrm{Tr}\,K\) (the rank of the projection) |
| Examples | GUE (sine kernel bulk / Airy edge), CUE (the circular Jacobi), Wishart (Laguerre / Bessel hard edge) |
The GUE kernel and the orthogonal polynomial method¶
The GUE eigenvalue density is a DPP with the kernel $$ K_N(x,y)=\sum_{k=0}^{N-1}\psi_k(x)\psi_k(y)\,e^{-(x^2+y^2)/2}, $$ the Hermite functions \(\psi_k\) (the orthogonal polynomials on \(\mathbb{R}\) with the Gaussian weight \(e^{-x^2}\)). The Christoffel–Darboux formula gives $$ K_N(x,y)=\frac{\psi_N(x)\psi_{N-1}(y)-\psi_{N-1}(x)\psi_N(y)}{x-y} $$ (the Hermite kernel). The Deift–Zhou steepest-descent Riemann–Hilbert method: the asymptotic analysis of the orthogonal polynomials / the kernel via the Riemann–Hilbert problem (the \(2\times2\) RH problem for the orthogonal polynomials), giving the sine kernel in the bulk and the Airy kernel at the edge (the Plancherel–Rotach asymptotics).
| GUE kernel data | Description |
|---|---|
| Kernel \(K_N\) | \(\sum_{k=0}^{N-1}\psi_k(x)\psi_k(y)e^{-(x^2+y^2)/2}\) (the Hermite functions) |
| Orthogonal polynomials | Hermite (Gaussian weight), Laguerre (Wishart / hard edge), Jacobi (the circular / Jacobi) |
| Christoffel–Darboux | \(K_N(x,y)=\frac{\psi_N\psi_{N-1}-\psi_{N-1}\psi_N}{x-y}\) (the rank-\(N\) kernel) |
| RH method | the Deift–Zhou steepest descent (the \(2\times2\) RH problem) |
| Limiting kernels | sine (bulk), Airy (soft edge), Bessel (hard edge) |
The Airy kernel and the Tracy–Widom distribution¶
The Airy kernel $$ K_{\mathrm{Ai}}(x,y)=\frac{\mathrm{Ai}(x)\mathrm{Ai}'(y)-\mathrm{Ai}'(x)\mathrm{Ai}(y)}{x-y}, $$ the edge scaling limit of the GUE kernel. The Airy point process is the DPP with the Airy kernel (the edge of the GUE). The Tracy–Widom distribution $$ F_2(s)=\det(1-K_{\mathrm{Ai}}|_{[s,\infty)})=\exp\Bigl(-\int_s^\infty (x-s)q(x)^2\,dx\Bigr), $$ the distribution of the largest point of the Airy process (the largest eigenvalue of the GUE), with \(q\) the Hastings–McLeod Painlevé II solution (\(q''=sq+2q^3\), \(q(s)\sim\mathrm{Ai}(s)\) as \(s\to+\infty\)).
| Airy / Tracy–Widom data | Description |
|---|---|
| Airy kernel \(K_{\mathrm{Ai}}\) | \(\frac{\mathrm{Ai}(x)\mathrm{Ai}'(y)-\mathrm{Ai}'(x)\mathrm{Ai}(y)}{x-y}\) |
| Airy process | the DPP with \(K_{\mathrm{Ai}}\) (the GUE edge) |
| Tracy–Widom \(F_\beta\) | the \(\beta=1,2,4\) versions (\(F_2\) the GUE, \(F_1\) the GOE, \(F_4\) the GSE) |
| Painlevé II | \(q''=sq+2q^3\), the Hastings–McLeod solution, \(F_2=\exp(-\int_s^\infty(x-s)q^2)\) |
| Edge universality | the \(F_\beta\) for all Wigner (the \(\beta=1,2,4\) edge universality) |
β-ensembles and loop equations¶
The \(\beta\)-ensemble (the general-\(\beta\) generalization, the log-gas with the \(\beta\) repulsion): $$ P(\lambda_1,\dots,\lambda_N)\propto\prod_{i<j}|\lambda_i-\lambda_j|^\beta\,e^{-N\sum V(\lambda_i)}. $$ The loop equations / Dyson–Schwinger equations: the integration-by-parts identities for the \(\beta\)-ensemble density, giving the recursion for the correlation functions and the free energy. The topological expansion (the \(1/N^2\) expansion of the free energy, the genus expansion, the 't Hooft / the matrix-model connection to mirror symmetry / GW).
| \(\beta\)-ensemble data | Description |
|---|---|
| \(\beta\)-ensemble | $\prod |
| Loop equations | the integration-by-parts / Dyson–Schwinger recursion for the correlators |
| Topological expansion | the \(1/N^2\) genus expansion (the 't Hooft / the matrix-model connection) |
| \(\beta\)-Tracy–Widom | the Ramírez–Rider–Virág stochastic Airy operator (the general-\(\beta\) edge) |
The \(\beta\)-Tracy–Widom (the general-\(\beta\) edge distribution): the Ramírez–Rider–Virág \(\beta\)-Tracy–Widom as the stochastic Airy operator — the Schrödinger operator \(-\frac{d^2}{dx^2}+x+\frac{2}{\sqrt{\beta}}B'(x)\) (the Airy operator plus the white-noise potential), with the \(\beta\)-Tracy–Widom the distribution of the smallest eigenvalue of the stochastic Airy operator.
See it¶
GUE kernel → Airy kernel → Tracy–Widom via the Painlevé II. The GUE \(k\)-point correlation \(\rho_k=\det(K_N(x_i,x_j))\) with the Hermite kernel \(K_N\). In the edge scaling \(\lambda_{\max}=2+N^{-2/3}s\), the kernel \(K_N\to K_{\mathrm{Ai}}\) (the Airy kernel, the Plancherel–Rotach / the Deift–Zhou steepest descent of the Hermite functions near the edge \(x=2\)). The largest-eigenvalue distribution: $$ P(\lambda_{\max}<2+N^{-2/3}s)\to F_2(s)=\det(1-K_{\mathrm{Ai}}|_{[s,\infty)}), $$ the Fredholm determinant of the Airy kernel on \([s,\infty)\). The Painlevé II: $$ F_2(s)=\exp\Bigl(-\int_s^\infty (x-s)q(x)^2\,dx\Bigr),\qquad q''=sq+2q^3,\quad q(s)\sim\mathrm{Ai}(s)\text{ as }s\to+\infty $$ (the Hastings–McLeod solution, the unique solution with the Airy asymptotics at \(+\infty\) and \(q(s)\sim\sqrt{-s/2}\) at \(-\infty\)). The key point: the GUE kernel → the Airy kernel → the Tracy–Widom via the Fredholm determinant, with the Painlevé II as the differential equation — the rigorous RMT computation of the edge distribution, the foundation for the Tracy–Widom universality.
Check¶
Verify that the Airy kernel \(K_{\mathrm{Ai}}\) is the edge limit of the GUE Hermite kernel \(K_N\), and identify the scaling.
The GUE kernel \(K_N(x,y)=\sum_{k=0}^{N-1}\psi_k(x)\psi_k(y)e^{-(x^2+y^2)/2}\) (the Hermite functions). Near the edge \(x\approx 2\) (the semicircle edge), the Plancherel–Rotach asymptotics of the Hermite functions \(\psi_k\) (the Airy asymptotics, the transition from the oscillatory bulk to the exponentially-decaying exterior) give, after the edge scaling \(x=2+N^{-2/3}s\), \(y=2+N^{-2/3}t\):
$$ K_N(2+N^{-2/3}s,\,2+N^{-2/3}t)\to N^{2/3}K_{\mathrm{Ai}}(s,t), $$ with \(K_{\mathrm{Ai}}(s,t)=\frac{\mathrm{Ai}(s)\mathrm{Ai}'(t)-\mathrm{Ai}'(s)\mathrm{Ai}(t)}{s-t}\) the Airy kernel. The \(N^{2/3}\) factor is the edge density \(\rho_{\mathrm{sc}}(2)\sim N^{2/3}\) (the square-root vanishing), and the rescaled kernel \(N^{-2/3}K_N\to K_{\mathrm{Ai}}\) is the edge DPP. So the Airy kernel is the edge scaling limit, the \(N^{-2/3}\) scaling the square-root edge density.
Show that the \(\beta\)-Tracy–Widom (the stochastic Airy operator) reduces to the Tracy–Widom \(F_2\) at \(\beta=2\).
The stochastic Airy operator \(H_\beta=-\frac{d^2}{dx^2}+x+\frac{2}{\sqrt{\beta}}B'(x)\) (the Airy operator plus the white noise). At \(\beta=2\), the coefficient \(2/\sqrt{\beta}=\sqrt{2}\), and the smallest eigenvalue of \(H_2\) has the distribution \(F_2\) (the Tracy–Widom, the Airy kernel Fredholm determinant). The connection: the \(\beta=2\) stochastic Airy operator is the edge limit of the GUE (the DBM / the soft edge), and the smallest eigenvalue is the largest GUE eigenvalue, the Tracy–Widom \(F_2\). For general \(\beta\), the stochastic Airy operator is the \(\beta\)-ensemble edge (the Ramírez–Rider–Virág theorem), giving the \(\beta\)-Tracy–Widom \(F_\beta\).
Pitfalls¶
- The DPP is for the GUE / CUE / Ginibre — the GOE / GSE are Pfaffian, not determinantal (a different structure: the correlation functions are Pfaffians of a \(2\times2\) matrix kernel, the quaternion determinant). Do not conflate the DPP (\(\beta=2\)) with the Pfaffian (\(\beta=1,4\)).
- The Tracy–Widom is the GUE edge (\(F_2\)); the GOE / GSE have \(F_1, F_4\) (the \(\beta=1,4\) versions, the Pfaffian / the Dyson edge), and the \(\beta\)-Tracy–Widom is the general \(\beta\) (the stochastic Airy operator, the Ramírez–Rider–Virág).
- The Deift–Zhou RH method is the asymptotic analysis of the orthogonal polynomials (the Hermite / Laguerre / Jacobi), not the exact computation — the RH gives the sine / Airy / Bessel asymptotics, the Fredholm determinant gives the Tracy–Widom.
- The topological expansion (the \(1/N^2\) genus expansion) is for the \(\beta\)-ensemble / the matrix model, the connection to the mirror symmetry / GW (the 't Hooft, the matrix-model / the topological string). It is a formal / asymptotic expansion, not a convergent one in general.
- The loop equations are the Dyson–Schwinger integration-by-parts identities; they give the recursion but the solvability / the topological expansion requires the spectral curve (the equilibrium measure \(\rho_V\)).
- The stochastic Airy operator is the \(\beta\)-ensemble edge (the Ramírez–Rider–Virág), not the DBM edge (the DBM is the \(\beta=1,2\) dynamics, the stochastic Airy is the general-\(\beta\) edge operator).
Refs¶
- Soshnikov, "Determinantal random point fields" (Russian Math. Surveys 2000); Borodin, "Determinantal point processes" (2011)
- Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach (AMS 1999); Deift–Zhou, "A steepest descent method for oscillatory Riemann–Hilbert problems" (1993)
- Tracy–Widom, "Level-spacing distributions and the Airy kernel" (CPAM 1994); Hastings–McLeod, "Painlevé II" (1980)
- Ramírez–Rider–Virág, "Beta ensembles and stochastic Airy operators" (2011); Eynard, Counting Surfaces (the topological expansion)
Next¶
RMT index · Foundations · Universality & Log-Gases · Spiked Models & Statistics · Non-Hermitian & Circular · SPDE / KPZ & universality — the KPZ universality / the Tracy–Widom / the Airy · Mirror symmetry / GW · Functional analysis