Matrix Bootstrap Techniques

• Matrix bootstrap is a nonperturbative framework that constrains matrix-model observables using exact Schwinger–Dyson equations and positivity conditions on moment and Gram matrices.
• It leverages convex semidefinite programming and nonlinear relaxations to derive rigorous bounds for single-trace and multi-trace correlators in Hermitian one-matrix, multi-matrix, and tensor models.
• The approach extends to thermal and quantum mechanical systems by incorporating the KMS condition and stationarity equations, yielding high-precision bounds even at finite N.

Matrix bootstrap is a nonperturbative program that constrains correlators of matrix integrals or matrix quantum mechanics by combining exact Schwinger–Dyson equations or stationarity equations with positivity of moment or Gram matrices; in thermal settings it also incorporates the KMS condition written as a convex matrix inequality involving a matrix logarithm (Laliberte et al., 18 Mar 2026, Lin et al., 28 Jul 2025, Cho et al., 2024). In current arXiv usage, the term most often refers to Hermitian one-matrix models, multi-matrix quantum mechanics, and related tensor models, at finite $N$ or in the planar limit, with the basic observables given by single-trace moments and, when factorization is not assumed, multi-trace moments (Laliberte et al., 18 Mar 2026, Lin et al., 28 Jul 2025).

1. Terminology and scope

The phrase “matrix bootstrap” is not uniform across the literature. In the matrix-model and matrix-quantum-mechanics literature, it denotes bootstrap constraints on moments such as $\langle \tr M^k\rangle$, $\langle \tr M^k\,\tr M^l\rangle$, or $\langle \tr X_I X_I\rangle$, obtained from loop equations, positivity, and related consistency conditions (Laliberte et al., 18 Mar 2026, Lin et al., 28 Jul 2025). In parts of the scattering-amplitude literature, however, “Matrix Bootstrap” is explicitly identified as a typo or shorthand for the S-matrix bootstrap, whose object is the scattering matrix $S$, constrained by analyticity, crossing, and unitarity rather than by matrix-model moment problems (Paulos et al., 2016, Paulos et al., 2016).

There is also a distinct statistical usage. “Matrix bootstrap” can denote constrained bootstrap testing of matrix rank, in which one studies minimum squared distances between an estimator and the manifold of fixed-rank matrix, and bootstraps the null law by resampling around a null-constrained estimator rather than around the unconstrained estimate (Portier et al., 2013). That usage is methodologically separate from the matrix-model and quantum-mechanical programs.

The physically central meaning in recent high-energy theory is therefore the bootstrap of matrix-model or matrix-quantum-mechanical observables. In that setting, the problem is to determine which collections of moments can arise from a consistent large-$N$ or finite-$N$ state, without solving the model by direct diagonalization or Monte Carlo sampling (Lin et al., 28 Jul 2025, Laliberte et al., 18 Mar 2026).

2. Algebraic foundation in one-matrix models

A canonical starting point is the Hermitian one-matrix quartic model

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$

with weighted trace

$\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$

Its basic observables are the single-trace moments

$m_k \equiv \big\langle \tr M^k \big\rangle,$

and the double-trace moments

$\langle \tr M^k\rangle$0

Choosing $\langle \tr M^k\rangle$1 in the Schwinger–Dyson equation gives the finite-$\langle \tr M^k\rangle$2 recursion

$\langle \tr M^k\rangle$3

while for the $\langle \tr M^k\rangle$4-symmetric quartic potential one sets odd moments to zero, $\langle \tr M^k\rangle$5 (Laliberte et al., 18 Mar 2026).

Positivity enters through expectation values of norms. For a basis $\langle \tr M^k\rangle$6, the single-trace Gram or Hankel matrix is

$\langle \tr M^k\rangle$7

the double-trace matrix is

$\langle \tr M^k\rangle$8

and positivity of traceless components yields

$\langle \tr M^k\rangle$9

The finite-$\langle \tr M^k\,\tr M^l\rangle$0 matrix bootstrap problem is then the simultaneous imposition of

$\langle \tr M^k\,\tr M^l\rangle$1

together with the Schwinger–Dyson recursion (Laliberte et al., 18 Mar 2026).

In the planar one-matrix problem, the same structure can be expressed through the resolvent

$\langle \tr M^k\,\tr M^l\rangle$2

which obeys

$\langle \tr M^k\,\tr M^l\rangle$3

with solution

$\langle \tr M^k\,\tr M^l\rangle$4

The corresponding spectral density satisfies

$\langle \tr M^k\,\tr M^l\rangle$5

For the Hermitian one-matrix model, the paper’s central theorem is

$\langle \tr M^k\,\tr M^l\rangle$6

where positivity of the resolvent means that the spectral density is supported on the real axis and has positive weight (Kazakov et al., 2021). This identifies exactly what the bootstrap positivity constraint is enforcing in the one-matrix case.

3. Finite-$\langle \tr M^k\,\tr M^l\rangle$7 one-matrix and tensor models

At finite $\langle \tr M^k\,\tr M^l\rangle$8, large-$\langle \tr M^k\,\tr M^l\rangle$9 factorization is not assumed, so $\langle \tr X_I X_I\rangle$0 in general. The finite-$\langle \tr X_I X_I\rangle$1 bootstrap therefore keeps the double-trace data as independent variables and treats the two-point function

$\langle \tr X_I X_I\rangle$2

as the primary bounded observable, equivalently determining the allowed region in the $\langle \tr X_I X_I\rangle$3-plane (Laliberte et al., 18 Mar 2026).

A striking result of this analysis is that, for the quartic one-matrix model and the constraints

$\langle \tr X_I X_I\rangle$4

“the resulting bounds on $\langle \tr X_I X_I\rangle$5 as a function of $\langle \tr X_I X_I\rangle$6 do not explicitly vary with $\langle \tr X_I X_I\rangle$7.” The interpretation offered is that the equations and positivity conditions, written in terms of the weighted-trace variables $\langle \tr X_I X_I\rangle$8, contain no explicit $\langle \tr X_I X_I\rangle$9, so finite-$S$0 information is encoded indirectly through the admissible structure of the multi-trace correlators (Laliberte et al., 18 Mar 2026).

This interpretation is supported by two limiting assumptions. Imposing large-$S$1-type factorization collapses the allowed region onto the planar branch, while imposing the $S$2 scalar-like identity

$S$3

recovers the $S$4 branch and excludes the large-$S$5 one. The paper therefore argues that genuinely $S$6-specific matrix-bootstrap bounds will likely require additional finite-dimensional algebraic structure, such as trace identities or Cayley–Hamilton-type relations (Laliberte et al., 18 Mar 2026).

The contrast with tensor models is direct. For the “pillow” tensor model, the Schwinger–Dyson equation contains an explicit factor $S$7, so “the tensor bootstrap produces genuinely $S$8-dependent bounds.” In this sense, the matrix case and the tensor case realize different finite-$S$9 mechanisms even when both are organized by Schwinger–Dyson equations and positivity (Laliberte et al., 18 Mar 2026).

4. Large-$N$0 matrix quantum mechanics

A second major branch of the subject studies $N$1-dimensional matrix quantum mechanics with $N$2 bosonic Hermitian traceless matrices $N$3 and conjugate momenta $N$4, governed by

$N$5

The bootstrap variables are single-trace expectations of matrix “words,” and the constraints come from cyclicity of the trace, Hermiticity, gauge invariance, global $N$6 symmetry, time-reversal and reflection/parity symmetries, positivity of operator norms, positivity of excitation energies above the ground state, and stationarity equations

$N$7

(Lin et al., 28 Jul 2025).

At $N$8, multi-trace correlators factorize, which closes the bootstrap on single-trace moments but introduces quadratic relations. The paper organizes operators by a level hierarchy,

$N$9

and complements the standard moment-matrix constraint

$N$0

with the ground-state positivity matrix

$N$1

The latter is emphasized as a major precision improvement (Lin et al., 28 Jul 2025).

To handle the nonlinearity from factorization, the paper uses nonlinear relaxation. Writing $N$2, one introduces a relaxed matrix $N$3 obeying

$N$4

This produces a convex semidefinite feasibility problem, solved numerically with SDPA-GMP (Lin et al., 28 Jul 2025).

The resulting bounds are exceptionally sharp. In the massive $N$5 model at level 14,

$N$6

and

$N$7

In the massless $N$8 case at level 14,

$N$9

For the bosonic BFSS case $$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$0, at level 11 the bounds are

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$1

The abstract summarizes the overall outcome as “more precise than large $$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$2, continuum extrapolations of lattice Monte Carlo simulations,” with some observables determined “up to 8 significant digits” (Lin et al., 28 Jul 2025).

5. Thermal and positivity-free extensions

The thermal bootstrap of matrix quantum mechanics extends the zero-temperature framework to the canonical state

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$3

for the ungauged one-matrix Hamiltonian

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$4

Here the genuinely thermal ingredient is the KMS condition, encoded through

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$5

and the matrix inequality

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$6

Because the matrix logarithm is nonpolynomial, the paper replaces it by a semidefinite relaxation of the relative-entropy cone, thereby obtaining an SDP that combines stationary state conditions, thermal inequalities, and semidefinite relaxations of matrix logarithm (Cho et al., 2024).

The main observable is the thermal energy

$$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$7

At large $$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$8, the upper and lower bounds on $$Z=\int dM\, e^{-N\,V(M)}, \qquad V(M)=\frac12 M^2+\frac g4 M^4,$$9 become extremely tight, with a relative gap of order $\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$0 at intermediate temperature, and the low-temperature behavior

$\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$1

is sharp enough to extract the lowest adjoint-sector gap $\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$2 within about $\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$3 from the lower bound and $\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$4 from the upper bound. In a negative-coupling metastable regime, SDP infeasibility is used to produce upper bounds on a thermal critical temperature (Cho et al., 2024).

A different extension removes positivity from the bootstrap altogether. For the large-$\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$5 Hermitian one-matrix model with action

$\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$6

the positivity-free method treats the bootstrap as a self-consistency problem between an eigenvalue distribution $\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$7 and the moments

$\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$8

The moments must satisfy the loop equations and simultaneously be generated by $\tr(\cdots)\equiv \frac1N \mathrm{Tr}(\cdots), \qquad \tr \mathbf{1}=1.$9, and the numerical problem is formulated as a least-squares minimization of

$m_k \equiv \big\langle \tr M^k \big\rangle,$0

The paper is explicit that this is “an approximate self-consistency bootstrap, not a rigorous bound-generating one,” but it reports that the method reproduces “with very high accuracy” the exact Euclidean solution and the perturbative Minkowski result, while avoiding a sign problem in principle (Maeta, 22 Jan 2026).

6. Significance, limitations, and relation to neighboring bootstrap programs

Taken together, these works define matrix bootstrap as a family of consistency programs for matrix models and matrix quantum mechanics. The common structure is the replacement of direct solution methods by moment constraints: Schwinger–Dyson equations in matrix integrals, $m_k \equiv \big\langle \tr M^k \big\rangle,$1 in quantum mechanics, positivity of moment or Gram matrices, and, in thermal problems, a KMS condition recast as a convex matrix inequality (Laliberte et al., 18 Mar 2026, Lin et al., 28 Jul 2025, Cho et al., 2024).

The main limitations are equally clear. In finite-$m_k \equiv \big\langle \tr M^k \big\rangle,$2 one-matrix models, the currently implemented constraints are “numerically insensitive to $m_k \equiv \big\langle \tr M^k \big\rangle,$3” unless extra finite-dimensional trace identities are added (Laliberte et al., 18 Mar 2026). In large-$m_k \equiv \big\langle \tr M^k \big\rangle,$4 multi-matrix quantum mechanics, the basis of words grows explosively, which is why $m_k \equiv \big\langle \tr M^k \big\rangle,$5 presently stops at level 11 while $m_k \equiv \big\langle \tr M^k \big\rangle,$6 reaches level 14 (Lin et al., 28 Jul 2025). In the thermal bootstrap, the real bottleneck is symbolic and memory growth, and the gauged case remains unresolved because the KMS constraint becomes trivial in the planar limit when only traced operators are allowed (Cho et al., 2024). In the positivity-free approach, the tradeoff for bypassing semidefinite positivity is ansatz dependence and loss of rigorous bounds (Maeta, 22 Jan 2026).

This matrix-model meaning should be sharply distinguished from the neighboring S-matrix bootstrap program. The 2016 foundational papers state that their subject is the S-matrix bootstrap of massive QFT, not random matrices or matrix models, and the associated white paper characterizes the modern program as numerically mapping out the space of allowed scattering amplitudes using analyticity, crossing, and unitarity (Paulos et al., 2016, Paulos et al., 2016, Kruczenski et al., 2022). The shared word “bootstrap” signals a common methodology of consistency-based constraint, but the two subjects act on different objects: matrix moments and operator algebras in one case, scattering amplitudes in the other.