Loop Equations in Matrix, Gauge & Tensor Models

• Loop equations are exact relations among observables derived from invariance principles that structure complete hierarchies of correlation functions.
• They apply across diverse settings—including matrix models, spectral curves, and tensor integrals—providing analytic tools for finite-N and asymptotic regimes.
• Their systematic use unifies methods in random matrix theory, gauge theory, and topological recursion, offering insights into universality and integrable systems.

Loop equations are exact relations among observables—typically resolvents, traces, or Wilson loops—obtained from integration by parts, infinitesimal changes of variables, or Haar-measure invariance. In different domains they appear as Schwinger–Dyson equations, Ward identities, Nekrasov equations, Makeenko–Migdal equations, or Virasoro/$\mathcal W$-constraints, but their common role is to organize full hierarchies of correlation functions rather than isolated moments (Eynard et al., 2016, Dimitrov et al., 2021, Perez-Sanchez, 2024). In modern usage, the term covers both finite-dimensional algebraic identities for matrix or tensor integrals and analytic hierarchies on spectral curves, differential systems, and microscopic random-matrix limits.

1. General definition and structural form

In random matrix theory, a basic source of loop equations is the eigenvalue measure

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$

whose polynomial moments satisfy linear constraints because total derivatives integrate to zero (Eynard, 2019). Writing $p_k=\sum_i x_i^k$ and $p_\mu=\prod_j p_{\mu_j}$, the one-matrix loop-equation generators are

$$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$

and they arise from the identity

$$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$

This is the archetypal Schwinger–Dyson mechanism: loop equations are the vanishing conditions $E(\mathcal L)=0$ for a distinguished subspace $\mathcal L\subset\mathcal P_N$ (Eynard, 2019).

A more geometric formulation replaces moments by multidifferentials $\omega_n^g$ near cuts of a local spectral curve. There, linear loop equations require the symmetric part

$\mathcal S f(z)=f(z)+(\iota^*f)(z)$

to extend holomorphically near the cuts, while quadratic loop equations require

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$0

to have double zeros at ramification points (Borot et al., 2013). This abstraction isolates the local analytic content common to matrix models, repulsive particle systems, loop models, knot invariants, and Liouville-type theories.

2. Finite-$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$1 matrix models and completeness of the equations

In finite-$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$2 matrix models, loop equations are not merely necessary constraints. For one-matrix models with polynomial $$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$3, every linear functional on symmetric polynomials that satisfies the loop equations is realized by integrating against an actual random-matrix eigenvalue measure on a suitable integration cycle (Eynard, 2019). The contour is not restricted to $$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$4: admissible complex Jordan arcs, or linear combinations of their homology classes, are required in order to capture all solutions. The admissible one-particle homology has dimension

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$5

and the $$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$6-particle space has

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$7

The theorem

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$8

identifies the full solution space of the loop equations with the homology of admissible contours (Eynard, 2019). A standard source of apparent nonuniqueness is therefore geometric rather than pathological: different contours correspond to different solutions.

For the arbitrary-$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$9 two-matrix model, exact loop equations still exist, but the leading spectral object is no longer an algebraic curve. The master loop equation takes the form

$p_k=\sum_i x_i^k$0

with

$p_k=\sum_i x_i^k$1

so the $p_k=\sum_i x_i^k$2-deformation promotes the classical spectral curve to a quantum spectral curve

$p_k=\sum_i x_i^k$3

(Bergère et al., 2011). In the completely degenerate sector, $p_k=\sum_i x_i^k$4, the loop equations reduce to Bethe-type equations

$p_k=\sum_i x_i^k$5

and higher corrections are computed by a matrix-valued topological recursion (Bergère et al., 2011).

3. Abstract loop equations, differential systems, and topological recursion

A decisive conceptual shift is that loop equations do not require an underlying matrix integral. For any differential system

$p_k=\sum_i x_i^k$6

with $p_k=\sum_i x_i^k$7 valued in a Lie group and $p_k=\sum_i x_i^k$8 a $p_k=\sum_i x_i^k$9-valued 1-form on a Riemann surface, one can define correlators $p_\mu=\prod_j p_{\mu_j}$0 from the flat transport kernel and the adjoint-flat section

$p_\mu=\prod_j p_{\mu_j}$1

and these correlators satisfy universal loop equations expressed through Lie-theoretic Casimirs and the characteristic polynomial of $p_\mu=\prod_j p_{\mu_j}$2 (Eynard et al., 2016). The master identity is

$p_\mu=\prod_j p_{\mu_j}$3

which realizes the same formal structure as matrix-model loop equations and $p_\mu=\prod_j p_{\mu_j}$4-algebra Ward identities (Eynard et al., 2016).

For $p_\mu=\prod_j p_{\mu_j}$5 traceless systems, equivalently scalar second-order ODEs, the Christoffel–Darboux kernel

$p_\mu=\prod_j p_{\mu_j}$6

generates determinantal correlators that satisfy the exact loop equations

$p_\mu=\prod_j p_{\mu_j}$7

with $p_\mu=\prod_j p_{\mu_j}$8 rational in $p_\mu=\prod_j p_{\mu_j}$9 and determined by the differential operator $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$0 (0901.3273). Under a topological $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$1-expansion, these exact identities feed directly into Eynard–Orantin topological recursion (0901.3273).

The abstract formulation makes this universality precise: if $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$2 satisfy solvable linear loop equations and quadratic loop equations, then they are given by the residue formula

$$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$3

which is exactly topological recursion (Borot et al., 2013). The same logic underlies the passage from local topological-recursion constraints to global Virasoro constraints for the Gromov–Witten theory of $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$4, where the logarithmic spectral datum

$$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$5

forces regularized open periods $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$6 in addition to Laurent-coefficient functionals $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$7 (Borot et al., 2019). In a different direction, the GUE partition function satisfies a Dubrovin–Zhang loop equation that coincides with the loop equation of the nonlinear Schrödinger Frobenius manifold, and

$$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$8

(Yang, 2024).

4. Microscopic loop equations and universal random-matrix statistics

At microscopic scale, loop equations can become complete characterization principles for limiting point processes. The recent theorem “Loop Equations Characterize Random Matrix Statistics” proves that for every rational $$\begin{aligned} Q_\mu &= \sum_{j=0}^{d} t_{j+1} p_{\mu_1+j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{j=0}^{\mu_1-1} p_j p_{\mu_1-1-j}p_{\mu_2}\dots p_{\mu_n} \ &\quad - \sum_{i=2}^n \mu_i \ p_{\mu_1+\mu_i-1} \prod_{k\neq j} p_{\mu_k}, \end{aligned}$$9, the $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$0 point process is the unique solution of the bulk loop equation hierarchy, and the $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$1 point process is the unique solution of the edge loop equation hierarchy (Bourgade et al., 8 Jul 2026). In this setting a point configuration $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$2 is encoded by a particle-generated Nevanlinna function

$$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$3

with $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$4. This representation is stable under local-uniform limits and treats bulk and edge in a unified way (Bourgade et al., 8 Jul 2026).

The bulk hierarchy is

$$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$5

and the edge hierarchy is the same identity with $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$6 replaced by $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$7 (Bourgade et al., 8 Jul 2026). The distinction reflects the difference between bulk $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$8-scaling and soft-edge $$Q_\mu(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} = \sum_{i=1}^N \frac{\partial}{\partial x_i} \left( x_i^{\mu_1}p_{\mu_2}(X)\cdots p_{\mu_n}(X)\Delta(X)^2 e^{-N\operatorname{Tr}V(X)} \right).$$9-scaling.

The corresponding uniqueness theorems are exact: $E(\mathcal L)=0$0 then the poles of $E(\mathcal L)=0$1 have the law of $E(\mathcal L)=0$2; and

$E(\mathcal L)=0$3

then the poles of $E(\mathcal L)=0$4 have the law of $E(\mathcal L)=0$5 (Bourgade et al., 8 Jul 2026). The proof linearizes the nonlinear hierarchy using exponential observables, producing deformed Calogero–Moser–Sutherland systems; establishes local laws such as

$E(\mathcal L)=0$6

in the bulk; analyzes the resulting holonomic PDE systems; and reconstructs the full family of resolvent correlation functions (Bourgade et al., 8 Jul 2026).

This yields a direct route to universality. If a microscopic ensemble satisfies approximate loop equations together with tightness and local moment bounds, then every subsequential limit satisfies the exact hierarchy, hence must be $E(\mathcal L)=0$7 or $E(\mathcal L)=0$8 (Bourgade et al., 8 Jul 2026). For Wigner matrices the approximate bulk loop equation is obtained from

$E(\mathcal L)=0$9

and the cumulant expansion

$\mathcal L\subset\mathcal P_N$0

while for random $\mathcal L\subset\mathcal P_N$1-regular graphs ordinary integration by parts is replaced by a discrete switching calculus (Bourgade et al., 8 Jul 2026). The paper also notes a sharp limitation: at $\mathcal L\subset\mathcal P_N$2 the bulk hierarchy degenerates and is satisfied by all homogeneous Poisson processes and their mixtures (Bourgade et al., 8 Jul 2026).

5. Multi-level, discrete, and generalized eigenvalue models

Loop equations extend beyond one-level eigenvalue gases. For $\mathcal L\subset\mathcal P_N$3-corners processes, the natural observables are the levelwise Stieltjes transforms

$\mathcal L\subset\mathcal P_N$4

and the loop equations become relations among joint cumulants across levels (Dimitrov et al., 2021). In the continuous setting, for the triangular interlacing array $\mathcal L\subset\mathcal P_N$5, the contour identity

$\mathcal L\subset\mathcal P_N$6

encodes the full multi-level Schwinger–Dyson hierarchy (Dimitrov et al., 2021). When $\mathcal L\subset\mathcal P_N$7, this reduces to the one-level Borot–Guionnet loop equations; the genuinely new feature is the appearance of adjacent-level combinations such as

$\mathcal L\subset\mathcal P_N$8

In the discrete setting, the same role is played by multi-level Nekrasov equations. For $\mathcal L\subset\mathcal P_N$9, analytic functions $\omega_n^g$0 and $\omega_n^g$1 are constructed from products of the form

$\omega_n^g$2

and for $\omega_n^g$3 these multiplicative observables collapse to additive differences of resolvents (Dimitrov et al., 2021). Deformation of the measure by factors

$\omega_n^g$4

turns derivatives with respect to $\omega_n^g$5 into joint cumulants, giving a systematic cumulant hierarchy (Dimitrov et al., 2021).

A different generalization replaces the Vandermonde interaction by an arbitrary difference-type measure,

$\omega_n^g$6

Defining

$\omega_n^g$7

one obtains the exact loop equation

$\omega_n^g$8

which is nonlocal because of the imaginary shifts in the spectral parameter (Vescovi et al., 2024). In the planar limit this framework yields an exact solution for the one-parameter $\omega_n^g$9-models, while at strong coupling it leads to a Wiener–Hopf/Riemann–Hilbert analysis with applications to $\mathcal S f(z)=f(z)+(\iota^*f)(z)$0 super-Yang–Mills theory and the Hoppe model (Vescovi et al., 2024).

6. Gauge theory, tensor models, quivers, and combinatorial formulations

In lattice Yang–Mills theory, loop equations govern Wilson loops

$\mathcal S f(z)=f(z)+(\iota^*f)(z)$1

through Schwinger–Dyson equations and trace relations (Liu et al., 7 Jan 2026). For SU(2), the basic trace relation

$\mathcal S f(z)=f(z)+(\iota^*f)(z)$2

becomes

$\mathcal S f(z)=f(z)+(\iota^*f)(z)$3

A central issue is the determination of a dynamically independent truncated basis for the lattice positivity bootstrap. The 2026 framework distinguishes direct equations, generated locally by plaquette-cut and subloop-cut algorithms, from indirect equations, which arise only after eliminating leaked higher-length loops (Liu et al., 7 Jan 2026). In 2D, for example, the truncation $\mathcal S f(z)=f(z)+(\iota^*f)(z)$4 contains 119 canonical loops and 51 direct equations, but enlarging to $\mathcal S f(z)=f(z)+(\iota^*f)(z)$5 yields 53 independent projected equations, exposing two hidden relations (Liu et al., 7 Jan 2026).

Tensor models admit analogous exact identities. Unitary invariants are indexed by colored bipartite bubbles, and reparametrization invariance yields tensor-model loop equations in which ordinary traces are replaced by bubble invariants (Krajewski et al., 2016). The formalism parallels matrix models, but the basic combinatorial objects are $\mathcal S f(z)=f(z)+(\iota^*f)(z)$6-bubbles rather than loops on a line or plane. In tensorial group field theory, the same invariants are lifted from finite tensors to fields $\mathcal S f(z)=f(z)+(\iota^*f)(z)$7 on $\mathcal S f(z)=f(z)+(\iota^*f)(z)$8 copies of a group, often with a closure constraint (Krajewski et al., 2016).

For quiver-based noncommutative geometries, a path integral over Dirac operators produces loop equations that generalize Makeenko–Migdal from lattices to arbitrary graphs (Perez-Sanchez, 2024). The spectral action

$\mathcal S f(z)=f(z)+(\iota^*f)(z)$9

is itself a sum over closed paths, and the exact quiver loop equations relate splitting of Wilson loops at a rooted edge to insertions of generalized plaquettes from the action (Perez-Sanchez, 2024). In the triangle-quiver example, the model reduces to the Gross–Witten–Wadia integral

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$00

providing an explicit check of the loop-equation-plus-positivity bootstrap (Perez-Sanchez, 2024).

Combinatorial and noncommutative master equations also arise in matrix models for random maps. For the 3-state Potts model coupled to 2D gravity, a single noncommutative generating equation

$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$01

packages the full planar hierarchy of loop equations for colored boundary words (Kulanthaivelu, 2019). Its coefficient extractions yield a closed system determining the fixed-spin spectral curve.

Taken together, these developments show that loop equations are not a single formal identity but a family of exact constraint hierarchies whose meaning depends on the observables and the ambient category: polynomial moments on contour homology classes, multidifferentials on local spectral curves, resolvents on interlacing arrays, Wilson loops on lattices or quivers, bubble observables in tensor models, or microscopic transforms of point processes. Their scope ranges from finite-$$d\mu(x_1,\dots,x_N)\propto \prod_{1\le i<j\le N}(x_i-x_j)^2 \prod_{i=1}^N e^{-V(x_i)}\,dx_i,$$02 reconstruction theorems to universality at microscopic scale, and from classical Schwinger–Dyson derivations to holonomic PDE systems, topological recursion, and bootstrap formulations (Eynard, 2019, Borot et al., 2013, Bourgade et al., 8 Jul 2026).