Constrained Dyson Brownian Motion

• Constrained Dyson Brownian Motion is a variant of Dyson’s eigenvalue diffusion characterized by interlacing constraints inherited from matrix minors.
• The process employs a singular covariance structure and Bessel-type repulsion to enforce geometric restrictions near the boundaries.
• It extends classical Dyson motion by maintaining Markov diffusion for two consecutive minors while highlighting non-Markovianity for additional layers.

Searching arXiv for the main paper and closely related work on constrained Dyson Brownian motion, interlacing minors, and geometric/ensemble-constrained variants. Constrained Dyson Brownian motion denotes variants of Dyson’s eigenvalue diffusion in which the particle system is restricted by additional geometric, algebraic, or state-space constraints beyond the standard non-collision condition. A canonical and concrete instance is the joint evolution of the spectra of two consecutive principal minors of a matrix undergoing Dyson’s Brownian motion: the two spectra must satisfy an interlacing relation, and their coupled dynamics form a Markov diffusion on the corresponding Gelfand–Tsetlin cone (Adler et al., 2010). In this setting, the constraint is neither an external hard wall nor an imposed reflection rule; it is inherited from matrix minor structure and is expressed simultaneously through the allowed state space, singular cross-covariances, and the stationary density (Adler et al., 2010).

1. Dyson Brownian motion and the meaning of constraint

Dyson’s original construction starts from matrix-valued Ornstein–Uhlenbeck dynamics on $H_n^{(\beta)}$, the space of real symmetric, complex Hermitian, or quaternionic self-dual Hermitian matrices for $\beta=1,2,4$. The independent real parameters of the matrix evolve as Ornstein–Uhlenbeck processes, and the invariant measure is the Gaussian ensemble $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$ (Adler et al., 2010). The ordered eigenvalues $\lambda_1(t),\dots,\lambda_n(t)$ then satisfy the Dyson SDE

$d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$

whose drift combines Ornstein–Uhlenbeck confinement with logarithmic repulsion (Adler et al., 2010).

In this basic form, Dyson Brownian motion already contains a constraint: eigenvalues are ordered and almost surely non-colliding. The more specific notion of constrained Dyson Brownian motion arises when further restrictions are imposed on the admissible configurations or on the geometry of the process. The most explicit example in the literature represented here is the interlacing constraint between the spectra of consecutive principal minors (Adler et al., 2010). Other works treat different forms of constraint, including confinement by an external potential $V$ in $\beta$-Dyson Brownian motion (Holcomb et al., 2017), killing on a subdomain in the colliding regime (Guillin et al., 11 Apr 2025), confinement to the unit circle in unitary Dyson Brownian motion (Bertucci et al., 23 Apr 2025), and confinement to a Jordan curve in the plane (Guskov et al., 5 Mar 2026). These are related but distinct mechanisms.

For the single-level Dyson process, the generator on eigenvalues is

$$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$

and the invariant density is the Gaussian $\beta$-ensemble

$$\pi_\lambda(\lambda)\,d\lambda = C_{n,\beta}^{-1}e^{-\frac12\sum_i\lambda_i^2} \prod_{i<j}|\lambda_j-\lambda_i|^\beta \,d\lambda$$

(Adler et al., 2010). Constrained variants modify this baseline by changing the support, the coupling structure, or both.

2. Interlacing constraints from consecutive principal minors

Let $\beta=1,2,4$0 denote the top-left $\beta=1,2,4$1 principal minor of a matrix $\beta=1,2,4$2 undergoing Dyson Ornstein–Uhlenbeck dynamics, and let $\beta=1,2,4$3 be the consecutive smaller principal minor. If

$\beta=1,2,4$4

then classical interlacing of a Hermitian matrix with its principal minor gives

$\beta=1,2,4$5

(Adler et al., 2010).

This defines the joint state space

$\beta=1,2,4$6

which is a two-row Gelfand–Tsetlin cone (Adler et al., 2010). The constraint is intrinsic: it is not added to the diffusion after the fact, but inherited from the algebraic relation between consecutive minors.

The central theorem of Adler, Nordenstam, and van Moerbeke is that the joint process

$\beta=1,2,4$7

is itself a Markov diffusion for $\beta=1,2,4$8, and the formulas extend to arbitrary $\beta=1,2,4$9 at the level of SDE and generator (Adler et al., 2010). This gives a fully explicit realization of Dyson Brownian motion under interlacing constraints.

The mixed Vandermonde structure is central. Writing

$e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$0

and

$e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$1

the interlacing condition controls the sign and singular structure of these factors (Adler et al., 2010). A plausible implication is that the geometry of the constrained state space is best understood through these mixed Vandermonde products rather than through a simple reflecting-boundary picture.

3. Two-level Markov diffusion: SDEs, generator, and boundary behavior

The lower level $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$2 evolves as an ordinary Dyson Ornstein–Uhlenbeck diffusion of size $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$3: $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$4 whereas the upper level $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$5 has the same deterministic Dyson drift at size $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$6, but its noise is coupled to $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$7 through the full bordered-matrix structure (Adler et al., 2010). The coupling enters through covariances, not through an extra drift term: $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$8 Thus the deterministic drift in $e^{-\frac{\beta}{2}\operatorname{Tr} B^2}\,dB$9 is pure Dyson at level $\lambda_1(t),\dots,\lambda_n(t)$0, independent of $\lambda_1(t),\dots,\lambda_n(t)$1, while the interlacing constraint is enforced stochastically through the quadratic variation structure (Adler et al., 2010).

At generator level, the coupled process decomposes as

$\lambda_1(t),\dots,\lambda_n(t)$2

with cross-term

$\lambda_1(t),\dots,\lambda_n(t)$3

(Adler et al., 2010). This singular second-order operator is the analytic expression of constrained Dyson motion for two consecutive minors.

The process does not hit the forbidden non-interlacing region. Near a boundary face, the gaps

$\lambda_1(t),\dots,\lambda_n(t)$4

behave like square-root diffusions with positive drift away from zero: $\lambda_1(t),\dots,\lambda_n(t)$5

$\lambda_1(t),\dots,\lambda_n(t)$6

with $\lambda_1(t),\dots,\lambda_n(t)$7 at the boundary (Adler et al., 2010). This excludes a simple reflection interpretation. The paper explicitly notes that this two-level diffusion is not Warren’s interlaced Dyson process, whose boundary mechanism is different (Adler et al., 2010).

A common misconception is that interlacing-constrained Dyson systems must be implemented by reflecting Brownian motions at the interlacing faces. The two-level minor process shows otherwise: the constraint may instead be encoded by a singular covariance structure and Bessel-type repulsion from the boundary (Adler et al., 2010).

4. Transition law, invariant measure, and the role of $\lambda_1(t),\dots,\lambda_n(t)$8

For one level, the transition density involves the Harish-Chandra/Itzykson–Zuber integral $\lambda_1(t),\dots,\lambda_n(t)$9 and the usual Vandermonde factor $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$0 (Adler et al., 2010). For two consecutive minors, the transition density is more elaborate and contains three distinct Vandermonde-type contributions: $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$1 together with a generalized HCIZ-type integral $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$2 and integrals over sphere variables that arise from angular degrees of freedom of the border vectors (Adler et al., 2010).

The invariant density is

$d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$3

(Adler et al., 2010). The support is still the interlacing domain, so the constraint survives even when the mixed factor has exponent zero.

The case $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$4 is special because

$d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$5

so the mixed interaction disappears from the stationary density, although not from the support or from the dynamics (Adler et al., 2010). This often invites overinterpretation. The disappearance of the cross Vandermonde factor at equilibrium does not mean the two levels decouple; the coupling remains present in the admissible state space and in the stochastic cross-variation.

The construction is explicit for $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$6, where a matrix model exists, and Corollary 3 extends the SDE and generator to arbitrary $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$7 by taking the same covariance structure and generator as definition (Adler et al., 2010). In that sense, constrained two-level Dyson diffusion has a genuine $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$8-version beyond the classical matrix cases.

A related but different notion of constrained Dyson motion appears in $d\lambda_i(t) = \left( -\lambda_i(t) + \sum_{j\neq i} \frac{1}{\lambda_i(t)-\lambda_j(t)} \right)dt + \sqrt{\frac{2}{\beta}\,db_{ii}(t), \qquad i=1,\dots,n,$9-Dyson Brownian motion with confining potential $V$0: $V$1 whose invariant law is the general $V$2-ensemble

$V$3

(Holcomb et al., 2017). Here the “constraint” is confinement by $V$4, rather than interlacing between levels.

5. Markovianity breaks at three consecutive minors

A sharp structural feature of the consecutive-minor construction is that two levels are Markovian but three consecutive levels are not, at least for $V$5 (Adler et al., 2010). If

$V$6

then for generic initial conditions the triple process is not Markovian (Adler et al., 2010).

The mechanism is generator-theoretic. If a projection $V$7 of a diffusion $V$8 is Markovian, then the full generator must map functions of $V$9 into functions of $\beta$0. For three consecutive minors, Adler, Nordenstam, and van Moerbeke exhibit matrix observables whose evolution under the full Dyson generator depends on information not recoverable from $\beta$1 alone (Adler et al., 2010). In the $\beta$2 case, the missing information can be seen explicitly through phases of off-diagonal entries that affect determinants but are not encoded in the three spectra (Adler et al., 2010).

This establishes a precise boundary for interlacing-constrained Markovian Dyson dynamics in the principal-minor setting: two consecutive rows of the Gelfand–Tsetlin pattern suffice, three do not (Adler et al., 2010). A plausible implication is that the angular variables eliminated successfully in the two-level reduction cease to be eliminable at the three-level stage.

This non-Markovianity is important conceptually because it prevents an overly naive extrapolation from two-level formulas to full interlacing arrays. The two-level process is explicit and Markovian; higher-level constrained dynamics require different constructions.

6. Related constrained variants and broader geometric formulations

The principal-minor model is one important realization of constrained Dyson Brownian motion, but the broader literature represented here shows that “constraint” appears in several mathematically distinct forms.

In tridiagonal models for $\beta$3-Dyson Brownian motion with confining potential $\beta$4, the eigenvalues follow the constrained log-gas SDE while the matrix is represented by a Jacobi diffusion whose entries evolve according to explicit SDEs derived through Lanczos tridiagonalization (Holcomb et al., 2017). For the quadratic potential $\beta$5, the eigenvalue distribution is “constrained to lie roughly in $\beta$6” and the large-$\beta$7 analysis describes the dynamics of bounded principal submatrices after centering by $\beta$8 (Holcomb et al., 2017). This is a matrix-level realization of confinement-constrained Dyson flow rather than an interlacing constraint.

A different constrained regime is Dyson Brownian motion with collisions and killing on a domain $\beta$9, where

$$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$0

and the process is defined via a multivalued SDE with interaction potential

$$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$1

on $$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$2 (Guillin et al., 11 Apr 2025). For $$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$3, collisions occur almost surely in finite time, yet the singular drift remains integrable through collisions and the killed process in $$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$4 admits a unique quasi-stationary distribution under suitable assumptions (Guillin et al., 11 Apr 2025). This is a constraint by killing rather than by interlacing.

Geometric constraints also arise on compact or curved configuration spaces. For unitary Dyson Brownian motion, the particles are eigenangles on the circle and satisfy

$$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$5

in the unitary scaling considered by Bertucci and Pesce (Bertucci et al., 23 Apr 2025). Here the constraint is that eigenvalues live on $$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$6, and the macroscopic limit is studied through a viscosity equation for the primitive of the empirical measure (Bertucci et al., 23 Apr 2025). Likewise, Dyson Brownian motion has been constructed on a rectifiable Jordan curve $$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$7, where the Gibbs density is

$$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$8

and the arc-length parametrization induces an SDE with drift

$$\mathcal{A}_\lambda = \sum_{i=1}^n\left( \frac1\beta \frac{\partial^2}{\partial\lambda_i^2} + \left(-\lambda_i+\sum_{j\neq i}\frac{1}{\lambda_i-\lambda_j}\right) \frac{\partial}{\partial\lambda_i} \right),$$9

(Guskov et al., 5 Mar 2026). This is a geometric confinement constraint on a nontrivial one-dimensional manifold.

A further geometric perspective appears in the extrinsic construction of Dyson Brownian motion via projection of Brownian motion onto tangent and normal directions of isospectral group orbits in $\beta$0 (Huang et al., 2022). In that formulation, $\beta$1 controls anisotropy between tangential and normal noise, and in the limit $\beta$2 the eigenvalues evolve deterministically by Coulomb repulsion while the isospectral orbits move by minus one half times mean curvature (Huang et al., 2022). This suggests that constrained Dyson motion can often be understood as stochastic dynamics near a foliation by group orbits.

7. Significance and conceptual interpretation

The two-consecutive-minor diffusion of Adler, Nordenstam, and van Moerbeke provides a particularly sharp and explicit example of constrained Dyson Brownian motion because the entire constrained structure is available in closed form: SDE, generator, transition density, and invariant measure (Adler et al., 2010). The process may be viewed as two Dyson diffusions whose admissible configurations lie in a two-row Gelfand–Tsetlin cone and whose fluctuations are coupled by a singular cross-operator (Adler et al., 2010).

Three complementary descriptions of the constraint are equivalent in this setting. First, there is state-space restriction: only interlacing configurations are allowed. Second, there is stochastic enforcement: cross-covariances become singular as interlacing faces are approached and induce Bessel-type repulsion from the boundary. Third, there is stationary encoding: the invariant measure carries levelwise Vandermonde factors together with a mixed power of $\beta$3 (Adler et al., 2010).

This suggests a general editorial shorthand, Editor’s term: “constraint by inherited geometry,” for those Dyson systems where the restriction is induced by an underlying matrix, group-orbit, or manifold structure rather than by externally imposed boundary conditions. In the principal-minor model, the inherited geometry is the interlacing of spectra of consecutive minors (Adler et al., 2010). In Jordan-curve and unitary models, it is the ambient curve or circle (Guskov et al., 5 Mar 2026, Bertucci et al., 23 Apr 2025). In tridiagonal and $\beta$4-constrained models, it is the confining potential and the associated $\beta$5-ensemble (Holcomb et al., 2017).

The sharp failure of Markovianity at three consecutive minors is equally significant. It shows that not every natural constrained spectral projection of matrix Dyson dynamics closes as a diffusion (Adler et al., 2010). This delineates the scope of explicit constrained Dyson theories and clarifies why two-level interlacing diffusions occupy a special place: they are rich enough to display nontrivial constrained geometry, yet simple enough to remain Markovian.

In summary, constrained Dyson Brownian motion is not a single model but a family of Dyson-type diffusions modified by structural restrictions. Among them, the spectra of two consecutive principal minors under Dyson Ornstein–Uhlenbeck dynamics furnish a paradigmatic finite-$\beta$6 example: a Markov diffusion on an interlacing cone, with pure Dyson drift on each level, singular noise coupling between levels, explicit transition laws, and a stationary density reflecting both Vandermonde repulsion and interlacing geometry (Adler et al., 2010).