Frozen Jacobi Process

• Frozen Jacobi process is the deterministic limit of Jacobi diffusions in the zero-temperature regime, where dynamics converge to ODEs governed by classical Jacobi polynomials.
• It exhibits scaling limits that lead to well-known distributions such as the Wigner semicircle and Marchenko–Pastur laws, bridging random matrix theory and free probability.
• The process highlights connections between integrable systems and eigenvalue crystallization, offering insights into fluctuation theory and the role of orthogonal polynomial roots in equilibrium.

The frozen Jacobi process refers to the deterministic limit of interacting diffusion processes—known as Jacobi processes—on configuration spaces constrained by root systems, especially within the so-called “freezing” or zero-temperature regime. This regime is realized as the inverse temperature parameter (or interaction strength) $\beta\to\infty$ (or coupling constant $K\to\infty$), concentrating the system on configurations that minimize a log-gas energy functional. The mean-field equilibria and dynamics in this regime are governed by classical Jacobi polynomials, and the empirical measures of particle locations, after appropriate rescaling, converge to Wigner semicircle and Marchenko–Pastur-type distributions, connected via free probability limits.

1. Definition and Dynamical Formulation

Consider the $N$-particle Jacobi process on the alcove $$A_N = \{ x \in \mathbb{R}^N : -1 \le x_1 < \cdots < x_N \le 1 \}$$, characterized by the following stochastic differential equations for $X_t = (X_{t,1},...,X_{t,N})$: $$dX_{t,i} = 2\sqrt{1 - X_{t,i}^2}\, dB_{t,i} + K \Big[ (p-q) - (p+q) X_{t,i} + 2\sum_{j\ne i} \frac{X_{t,i} - X_{t,j}}{1 - X_{t,i} X_{t,j}} \Big] dt,$$ where $B_{t,i}$ are independent Brownian motions and

$$K = k_3 > 0,\quad p = N - 1 + \frac{1+2k_2}{2k_3},\quad q = N - 1 + \frac{1+2k_1+2k_2}{2k_3},$$

for parameters $k_1, k_2 \in \mathbb{R}$ ($k_2\ge0$, $k_1+k_2\ge0$). The process possesses singular logarithmic repulsion and a quadratic confining drift.

In the freezing limit $K\to\infty$ ($k_3\to\infty$), the stochastic term vanishes under suitable scaling, and the system reduces to a deterministic ODE on $(-1,1)^N$: $$\frac{d}{dt}x_i(t) = (p-q) - (p+q)x_i(t) + 2\sum_{j\ne i} \frac{x_i(t) - x_j(t)}{1 - x_i(t)x_j(t)},\quad i=1,...,N.$$ This deterministic evolution, which will be called the “frozen Jacobi process” (Editor's term), constitutes the central object of analysis.

2. Equilibria and the Role of Jacobi Polynomials

The unique stationary point of the frozen dynamics is given by the configuration of ordered zeros of the $N$th-degree Jacobi polynomial,

$$P_N^{(\alpha,\beta)}(x),\quad \alpha = q-N,\; \beta = p-N.$$

Explicitly, if $z = (z_1 < \cdots < z_N)$ are the zeros of $P_N^{(\alpha,\beta)}(x)$, then $z$ is the unique fixed point of the deterministic frozen Jacobi ODE. This result follows from a gradient flow argument using the trigonometric change of coordinates $x_i = \cos\theta_i$ and the associated log-gas potential.

This universality links the equilibrium of high-coupling Jacobi systems with the roots of orthogonal polynomials, a feature shared with other $\beta$-ensembles in random matrix theory.

3. Fluctuation Theory and Central Limit Regimes

For the Jacobi $\beta$-ensemble with joint probability density

$$p_{N,\alpha,\beta,\kappa}(x) = C_{N,\alpha,\beta,\kappa}\, \prod_{i<j} |x_j-x_i|^{\kappa} \prod_{i=1}^N (1-x_i)^{\alpha}(1+x_i)^{\beta},\quad x\in A_N,$$

as $\kappa\to\infty$ (“zero temperature”), the configuration concentrates at

$$Z_N^{(\alpha,\beta)} = (z_{1,N}^{(\alpha,\beta)},\dots,z_{N,N}^{(\alpha,\beta)}),$$

with $z_{i,N}^{(\alpha,\beta)}$ as above. Fluctuations about these points are governed, for large but finite $\kappa$, by a CLT: $$\sqrt{\kappa}\, \left(X^{(\kappa)} - Z_N^{(\alpha,\beta)}\right) \xrightarrow{d} N(0,E_N^{(\alpha,\beta)}),$$ where the inverse covariance $S_N=E_N^{-1}$ is given by

$$S_{ii} = \sum_{\ell\ne i} \frac{1}{(\mu_i-\mu_\ell)^2} + \frac{\alpha+1}{2(1-\mu_i)^2} + \frac{\beta+1}{2(1+\mu_i)^2}, \qquad S_{ij} = -\frac{1}{(\mu_i-\mu_j)^2},\quad i\ne j.$$

A spectral decomposition of the covariance uses dual orthogonal polynomials (in the sense of de Boor–Saff) supported on the zero set.

For the extremal eigenvalues (e.g., largest and smallest $r$), the $r$-dimensional principal submatrix of $E_N$ gives a joint Gaussian limit. In the large-$N$ hard-edge regime, these covariances are expressed, for fixed $r,s$, as

$$\lim_{N\to\infty} N^2\, \mathrm{Cov}(\lambda_{N-r+1}, \lambda_{N-s+1}) = \frac{1}{J_{\alpha+1}(j_{\alpha,r})\,J_{\alpha+1}(j_{\alpha,s})} \int_0^1 \frac{J_\alpha(j_{\alpha,r}\sqrt{y})J_\alpha(j_{\alpha,s}\sqrt{y})}{\sqrt{1-y}}dy,$$

with $j_{\alpha,r}$ the $r$th positive zero of the Bessel function $J_\alpha$ (Hermann et al., 3 Feb 2025).

4. Large-N Scaling, Empirical Distributions, and Limiting Laws

The empirical measure,

$$\mu_N(t) = \frac{1}{N}\sum_{i=1}^N \delta_{a_N(x_i(t)-b_N)},$$

under suitable affine scaling, converges as $N\to\infty$ to deterministic limiting laws, depending on the regime:

• Bulk regime (Wigner semicircle scaling): If $p_N/N \to p>0$, $q_N/N\to q>0$, set $b_N = (p_N-q_N)/(p_N+q_N)$, $a_N = \sqrt{p_N+q_N}$. The empirical measure converges to a semicircle law with radius modified by the free convolution with the initial data.
• Hard-edge regime (Marchenko–Pastur scaling): For $p_N/N\to p > 0$ and $q_N\to\infty$, set $b_N = \pm1$ and $a_N = q_N/N$. The limiting law is a Marchenko–Pastur law associated with the rescaled edge behavior.

These scaling limits yield almost sure convergence of empirical distributions and connect the frozen Jacobi process with central objects in free probability (semicircle, Marchenko–Pastur laws), where the limit is expressed in terms of free additive convolution, e.g.,

$\mu(t) = (e^{-t}\mu_0)\boxplus\mathrm{Sc}(R),$

with $\mathrm{Sc}(R)$ the semicircle law of radius $R$.

5. Finite-Free Transforms, Characteristic Polynomials, and Connection to Free Probability

The dynamics of the roots of the averaged characteristic polynomial,

$\chi_t(x) = \prod_{j=1}^m (x - x_j(t)),$

under the inverse Jacobi heat flow, define the frozen Jacobi process at finite $m$ (Demni et al., 4 Nov 2025). The ODE system governing the root dynamics is: $$\frac{d}{dt} x_j(t) = (p - (p+q)x_j(t)) + \sum_{k\ne j} \frac{x_j(t)(1-x_k(t)) + x_k(t)(1-x_j(t))}{x_j(t)-x_k(t)},$$ equivalently,

$$\frac{dx_j}{dt} = (r+1) - (r+s+2)x_j + 2x_j(1-x_j) \sum_{k\ne j} \frac{1}{x_j - x_k}.$$

This system is linked to finite-free $S$- and $T$-transforms. The latter satisfies, in the large-$m$ limit,

$$\partial_t T(t,z) = [2(1-z)\lambda\theta-1]T(t,z) + \theta[1-2\lambda(1-z)] + \theta(1-\lambda(1-z))(1-z)\partial_z \ln T(t,z),$$

which, under suitable identifications, gives the PDE for the S-transform of the free Jacobi process. As $m\to\infty$, the law of the frozen roots converges weakly to that of the free Jacobi process at each time (Demni et al., 4 Nov 2025).

An expansion of the characteristic polynomial in the Jacobi basis is given by

$$\chi_t^{(r,s,m)}(x) = \sum_{j=0}^m C_j^{(r,s,m)}\, e^{-(m-j)(r+s+m+j+1)t} Q_j^{(r,s)}(x),$$

where $Q_j^{(r,s)}$ are normalized Jacobi polynomials and $C_j^{(r,s,m)}$ explicit combinatorial coefficients.

6. Special Geometric Alignments and Symmetries

A notable feature arises in the parameter regime $(r,s)=(-1/2, -1/2)$ (hence $\lambda=1$, $\theta=1/2$), linking the Jacobi process to the unitary Hermite process via a Szegö variable transformation: $$H_{2m}(z,2t) = 4^m z^m \chi_t^{(-1/2,-1/2,m)}\left( \frac{z+z^{-1}+2}{4} \right ),$$ where the $H_{2m}$ are Hermite Unitary polynomials. The zeros of $H_{2m}$ on the unit circle, when pulled back under this mapping, correspond to the roots of the frozen Jacobi ODE. The resulting ODE for the angles of the zeros becomes the well-known frozen $\beta$-Dyson (type A) system on the circle.

7. Connections and Asymptotic Landscape

The frozen Jacobi process bridges integrable stochastic differential equations, classical log-gas models, and the field of free probability. For large particle number and different scaling regimes, it dynamically realizes Wigner semicircle and Marchenko–Pastur-type limiting distributions, with the initial condition entering via free convolution. The fluctuation theory sharpens to finite-rank Gaussian corrections and, at the hard edge, relates to Bessel kernel asymptotics for extremal eigenvalues (Auer et al., 2022, Hermann et al., 3 Feb 2025, Demni et al., 4 Nov 2025).

A plausible implication is that the dynamical perspective provided by the frozen Jacobi process furnishes a robust framework for analyzing the crystallization of eigenvalues in integrable interacting particle systems, strengthening the connections across random matrix theory, orthogonal polynomial theory, and free probability.