Symmetric Dyson Exclusion Process (SDEP)

• SDEP is a discrete, maximal-entropy exclusion process built on a ring and defined via a sine–Vandermonde ground state, ensuring non-collision through a Doob transform.
• It exhibits exact solvability using Schur functions, character theory, and free fermion mappings that yield explicit spectral properties and eigenvalue formulations.
• Its macroscopic behavior spans low-density convergence to Dyson Brownian Motion and finite-density non-local hydrodynamics, linking microscopic dynamics to continuum transport.

Searching arXiv for the cited papers to ground the article in current arXiv metadata. Using the arXiv search tool for (Offret, 4 Mar 2026). Using the arXiv search tool for (Zahra et al., 13 Aug 2025). The Symmetric Dyson Exclusion Process (SDEP) is a symmetric exclusion dynamics on a discrete ring in which particles execute nearest-neighbor jumps under hard-core exclusion, but the transition law is modified by a maximal-entropy or ground-state Doob transform built from a sine–Vandermonde eigenfunction. In the 2026 formulation, SDEP is precisely the Maximal Entropy Simple Symmetric Exclusion Process (MESSEP), the unique maximal-entropy Markov chain on the configuration graph of allowed single-particle nearest-neighbor jumps under exclusion; in the 2025 formulation, it is an exact continuous-time Doob transform of the spin-$1/2$ XX chain with reversible Dyson log-gas measure. These formulations place SDEP at the intersection of exclusion processes, random matrix theory, Schur and character theory, and free probability: at low density it converges to Unitary Dyson Brownian Motion (UDBM), while at finite density it gives non-local hydrodynamics on the circle and, as occupancy vanishes, the hydrodynamic equation of Free Unitary Brownian Motion (FUBM) (Offret, 4 Mar 2026, Zahra et al., 13 Aug 2025).

1. Microscopic definition and maximal-entropy structure

In the MESSEP formulation, the underlying space is the discrete ring $T_L=\{0,\dots,L-1\}$ with periodic edges, and the $N$-particle state space is

$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$

the set of unordered $N$-site subsets. Two configurations are adjacent if one particle moves by $\pm1$ to an empty site. Let $A$ be the symmetric adjacency matrix of this configuration graph, let $\rho$ be its spectral radius, and let $\psi>0$ satisfy $A\psi=\rho\psi$. The maximal-entropy transition kernel is the Doob transform

$T_L=\{0,\dots,L-1\}$0

with reversible invariant measure $T_L=\{0,\dots,L-1\}$1. This chain is ergodic for fixed $T_L=\{0,\dots,L-1\}$2, and among Markov chains compatible with the configuration graph it uniquely maximizes the asymptotic path entropy rate (Offret, 4 Mar 2026).

The Perron–Frobenius eigenfunction is explicit. Writing $T_L=\{0,\dots,L-1\}$3 for odd $T_L=\{0,\dots,L-1\}$4 and $T_L=\{0,\dots,L-1\}$5 for even $T_L=\{0,\dots,L-1\}$6, one obtains the sine–Vandermonde

$T_L=\{0,\dots,L-1\}$7

and

$T_L=\{0,\dots,L-1\}$8

The process coincides with simple random walk on $T_L=\{0,\dots,L-1\}$9 conditioned never to collide; equivalently, it is a system of independent symmetric random walks with exclusion enforced by a ground-state transform. This non-collision picture is central because the effective repulsion is not inserted ad hoc: it is generated by conditioning (Offret, 4 Mar 2026).

The continuous-time SDEP of the 2025 work uses the same sine–Vandermonde structure in rate form. On a ring of $N$0 sites, the ordered configuration $N$1 evolves by nearest-neighbor jumps with rates

$N$2

provided the target site is empty. The reversible invariant law is the discrete Dyson circular Coulomb gas

$N$3

This makes clear that the jumps remain nearest-neighbor, whereas the interaction is configuration-wide through the multiplicative sine ratios. A common misconception is therefore to identify SDEP with a local exclusion process solely because its move set is local; the defining non-locality lies in the rates or, equivalently, in the Doob transform (Zahra et al., 13 Aug 2025).

2. Exact solvability: Schur functions, characters, and free fermions

A distinctive feature of MESSEP is that its full spectral theory is explicit. For partitions $N$4 with $N$5, the Schur polynomial

$N$6

specialized at $N$7-th roots of unity yields an eigenfunction of $N$8. The collection

$N$9

forms an orthonormal eigenbasis in $$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$0. Under the identification of configurations with shifted partitions, these eigenfunctions coincide, up to phase, with $$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$1. The associated eigenvalues are

$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$2

The character-theoretic mechanism is encoded by the Frobenius identities relating power sums and Schur functions, with hook partitions $$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$3 playing a special role through Murnaghan–Nakayama. These hook expansions control moment asymptotics and the passage from discrete spectral sums to macroscopic transport equations (Offret, 4 Mar 2026).

The same process also admits a quantum-chain realization. In the 2025 formulation, the continuous-time generator is the exact ground-state transform of the spin-$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$4 XX Hamiltonian

$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$5

which Jordan–Wigner maps to free fermions:

$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$6

In the $$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$7-particle sector the ground state is

$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$8

and the Markov generator is

$$C_{L,N}=P_N(T_L)\simeq \{\xi=(\xi_1<\cdots<\xi_N)\in T_L^N\},$$9

The ground-state energy and spectral gap are

$N$0

This exact free-fermion representation supplies microscopic control over the spectrum, correlators, and equilibrium activity, while the Schur basis in the 2026 work provides the algebraic machinery for rigorous scaling limits. Together they show that SDEP is integrable in two complementary senses: through symmetric-function diagonalization and through fermionic linearization (Zahra et al., 13 Aug 2025).

3. Low-density scaling and the Dyson diffusion limit

When $N$1 is fixed and $N$2, the MESSEP converges after diffusive rescaling to UDBM on the unit circle. If $N$3 denotes the discrete position and

$N$4

then under $N$5 one has

$N$6

where the limiting eigenangles satisfy

$N$7

In the normalization used in the paper,

$N$8

The limiting generator has the ground-state form

$N$9

with

$\pm1$0

Convergence holds functionally in $\pm1$1, with tightness obtained through martingale estimates and Schur-polynomial expansions (Offret, 4 Mar 2026).

The physical interpretation is that the Dyson cotangent interaction emerges as an entropic force. Because MESSEP is simple random walk conditioned never to collide, the discrete drift is $\pm1$2; in the scaling limit this becomes $\pm1$3, which is precisely the Coulombic cotangent repulsion on the circle. The repulsion is therefore not a separate microscopic potential but the continuum imprint of exclusion under maximal-entropy conditioning (Offret, 4 Mar 2026).

Elementary cases already display the mechanism. For $\pm1$4, $\pm1$5 is constant, so MESSEP is simple random walk on the ring and the limit is ordinary Brownian motion on the circle. For $\pm1$6,

$\pm1$7

hence the invariant measure is proportional to $\pm1$8 and nearby pairs are entropically suppressed. At fixed time, the gap process is determinantal with Dirichlet kernel

$\pm1$9

which approaches the sine kernel as $A$0 with $A$1 (Offret, 4 Mar 2026).

4. Finite-density hydrodynamics and non-local transport

In the hydrodynamic regime $A$2, the empirical measure on the circle,

$A$3

converges to $A$4, with $A$5. The limiting density solves

$A$6

where $A$7 is the circular Hilbert transform

$A$8

The PDE holds globally in a weak sense and, under the stated non-saturation bounds, in the strong sense for all $A$9; more generally, there exists $\rho$0 such that $\rho$1 is analytic on $\rho$2 circle and converges exponentially to $\rho$3 (Offret, 4 Mar 2026).

The same hydrodynamic limit admits an analytic encoding through moments. If

$\rho$4

then $\rho$5 satisfies the complex Burgers-type equation

$\rho$6

with

$\rho$7

The method of characteristics gives

$\rho$8

where $\rho$9 is determined by

$\psi>0$0

and the boundary relation

$\psi>0$1

connects the analytic and real-variable formulations. In this representation, the non-local flux is the real part of a sine applied to the complexified density $\psi>0$2 (Offret, 4 Mar 2026).

A closely related, but explicitly conjectural, finite-density hydrodynamics appears in the continuous-time SDEP work:

$\psi>0$3

together with the equivalent local two-field system for $\psi>0$4, where $\psi>0$5, and the analytic equation

$\psi>0$6

This formulation yields the implicit solution

$\psi>0$7

Both papers therefore assign the macroscopic current a genuinely non-local dependence on the density through a Hilbert transform and a sine–sinh structure. This suggests a common constitutive mechanism inherited from the same sine–Vandermonde ground state, even though the exact scaling regimes and normalizations are not identical (Zahra et al., 13 Aug 2025).

5. Vanishing occupancy and free unitary Brownian hydrodynamics

The limit $\psi>0$8 of the MESSEP hydrodynamic equation recovers the standard non-local continuity law associated with logarithmic interaction on the circle. Using

$\psi>0$9

and

$A\psi=\rho\psi$0

one obtains

$A\psi=\rho\psi$1

According to the 2026 analysis, this is the hydrodynamic PDE governing the spectral measure of FUBM. On the free-probability side, if $A\psi=\rho\psi$2 is the spectral measure of large-$A\psi=\rho\psi$3 unitary Brownian motion, its Herglotz transform satisfies

$A\psi=\rho\psi$4

with initial $A\psi=\rho\psi$5; after translating between boundary values and real-variable densities, the MESSEP vanishing-density limit matches this free unitary setting (Offret, 4 Mar 2026).

The 2025 paper reaches the same low-density Dyson-gas structure from a different direction. For $A\psi=\rho\psi$6,

$A\psi=\rho\psi$7

so that

$A\psi=\rho\psi$8

leading to

$A\psi=\rho\psi$9

This is presented there as the known hydrodynamics of the continuous Dyson gas. The combined picture is that SDEP/MESSEP supplies a discrete entropic route to two continuum regimes: few-particle dynamics converging to UDBM, and vanishing-occupancy collective dynamics converging to free-unitary or Dyson-gas hydrodynamics, depending on the representation used (Zahra et al., 13 Aug 2025).

6. Comparison with other exclusion processes, explicit solutions, and unresolved issues

SDEP differs sharply from classical SSEP. In the 2026 comparison, continuous-time SSEP has diffusive hydrodynamics governed by the heat equation under standard scaling, whereas MESSEP exhibits an $T_L=\{0,\dots,L-1\}$00 scale with non-local transport. In the 2025 comparison, SSEP has diffusive scaling with local constitutive law, while SDEP has Eulerian $T_L=\{0,\dots,L-1\}$01 scaling and a current functional depending on the full profile through the Hilbert transform. Both accounts agree on the structural distinction: maximal entropy and symmetric adjacency select the sine–Vandermonde ground state and thereby generate Dyson-type repulsion and non-local hydrodynamics absent in standard exclusion models (Offret, 4 Mar 2026, Zahra et al., 13 Aug 2025).

The 2025 work also develops explicit hydrodynamic solutions for block initial data. For a single block on the ring,

$T_L=\{0,\dots,L-1\}$02

and in the rescaled infinite-line limit the implicit relation for $T_L=\{0,\dots,L-1\}$03 becomes a cubic equation whose discriminant

$T_L=\{0,\dots,L-1\}$04

defines the arctic curve. For two blocks, one obtains a higher-degree polynomial relation and a second discriminantal arctic curve. In both cases the asymptotic front is $T_L=\{0,\dots,L-1\}$05 and the density approaches the diffusive semicircle

$T_L=\{0,\dots,L-1\}$06

These limit shapes agree with large-scale Monte Carlo simulations, and small perturbations around uniform density relax with dispersion relation

$T_L=\{0,\dots,L-1\}$07

consistent with ballistic relaxation in that formulation (Zahra et al., 13 Aug 2025).

The literature currently contains both rigorous and conjectural components. The exact definition of the rates, the Doob transform, the reversible Dyson measure, and the XX/free-fermion mapping are exact in the 2025 work, but its hydrodynamic closure is explicitly described as conjectural. By contrast, the 2026 work proves low-density convergence to UDBM and a hydrodynamic limit for the empirical measure, with Schur and character theory furnishing the limit passage. A plausible implication is that the term “SDEP” is being used for closely related, but differently normalized or differently timed, Doob-transformed exclusion dynamics. The cited works do not provide a complete reconciliation of the ballistic $T_L=\{0,\dots,L-1\}$08-scale conjecture and the diffusive $T_L=\{0,\dots,L-1\}$09-scale theorem, so that point remains a natural locus for further analysis. Other open directions recorded in the 2025 study include coupling to reservoirs, boundary-driven phase transitions, large deviations and macroscopic fluctuation theory, asymmetry and KPZ-like corrections with Hilbert kernels, and extension from XX to XXZ-type activity tilts (Zahra et al., 13 Aug 2025).

In this sense, SDEP is best viewed not as a single phenomenological lattice gas but as a canonical entropic exclusion framework in which the sine–Vandermonde ground state organizes microscopic reversibility, exact solvability, Dyson repulsion, and non-local hydrodynamics. The central algebraic content is Schur and character theory; the central probabilistic content is the Doob conditioning against collisions; and the central continuum outputs are UDBM and free unitary hydrodynamics (Offret, 4 Mar 2026).