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Analytic and Numerical Bootstrap for One-Matrix Model and "Unsolvable" Two-Matrix Model

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Vladimir Kazakov, Zechuan Zheng

High Energy Physics - Theory · Submitted 10 Aug 2021 · 55 references

Pith's verdict: The one-matrix positivity theorem in Section 3.1 is a real, citable result; the two-matrix six-digit numbers are honest outer bounds whose tightness rests on an unmeasured relaxation gap.

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arXiv ID2108.04830 (version 3)
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CitationVladimir Kazakov, Zechuan Zheng. "Analytic and Numerical Bootstrap for One-Matrix Model and "Unsolvable" Two-Matrix Model." arXiv:2108.04830 (2021). https://pith.science/paper/W2X2T7YS
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The paper

The paper claims

The authors argue that the bootstrap method — exact loop equations plus positivity of the matrix of moments — can be turned into a rigorous and practical numerical tool for large-$N$ matrix integrals that resist analytic solution, and they背

The reading

Large-$N$ matrix integrals with two or more matrices are generally written off as analytically unsolvable, leaving Monte Carlo as the only general numerical tool, with three or four digits of precision and a statistical error bar. This paper asks whether the bootstrap idea — impose the exact loop (Schwinger-Dyson) equations on single-trace moments, then impose positivity of every inner product of operators — can be made into a practical and rigorous numerical method for such integrals. The authors answer in two parts. For the one-matrix model they prove that positivity of the correlation matrix of moments is exactly equivalent to the eigenvalue density being supported on the real axis with positive weight, which both explains why the bootstrap converges and renders the bootstrap problem analytically solvable. For multi-matrix models they replace the quadratic loop equations by linear equations plus a positivity condition on an auxiliary relaxation matrix, converting a non-convex problem into a standard semidefinite program that can be run at large cutoff. On a two-matrix model with a commutator-squared interaction and quartic potentials, the resulting bounds tighten to roughly six digits.

What carries the argument

Two objects carry the argument. The first is the correlation matrix $M_{ij}=langlemathrm{Tr},O_i^{dagger}O_jrangle$ over all words up to a length cutoff; its positive semidefiniteness is the only physical input beyond the loop equations, and the Hamburger moment theorem converts it into a statement about where the eigenvalue density lives. The second is the relaxation matrix, a bordered matrix built from the vector $x$ of moments and an independent matrix $X$ standing in for the products $x x^{T}$; imposing $begin{pmatrix}1&x^{T} x&Xend{pmatrix}succeq 0$ together with the loop equations linearised in $X$ yields a convex program whose optimum bounds any chosen moment from above and below. The relaxation matrix is what makes the problem solvable at large cutoff: it trades exact quadratic constraints for a much larger positive-semidefinite condition.

Extended reading notes

Core claim

The central claim is that matrix bootstrap can be made both rigorous and efficient. Rigour comes from a theorem: for the large-$N$ Hermitian one-matrix model, positive semidefiniteness of the Hankel matrix of moments $W_{ij}=W_{i+j-2}$ holds if and only if the resolvent has cuts only on the real axis with positive discontinuity, i.e. the eigenvalue density is a positive measure on the real line. This is proved by applying the solution of the Hamburger moment problem together with the exponential growth bound on moments implied by the explicit square-root form of the resolvent. The authors use this equivalence to solve the bootstrap conditions in closed form for the quartic potential $V(x)=tfrac{1}{2}mu x^2+tfrac{1}{4}g x^4$, classifying all admissible solutions: for $mu=1$ a unique $Z_2$-symmetric solution exists only for $gge -1/12$; for $mu=-1$ and $0<g<1/4$ the physical solutions form a one-parameter curve in $(W_1,W_2)$ with an explicit critical value $W_{1c}$, so the exact solution set is non-convex. Efficiency comes from the relaxation step: writing the quadratic terms of the loop equations as an independent symmetric matrix $X$ and weakening $X=xx^{T}$ to $Xsucceq xx^{T}$ (equivalently a bordered matrix $begin{pmatrix}1&x^{T} x&Xend{pmatrix}succeq 0$) turns the nonlinear SDP into a linear one. Applied to $Z=int dA,dB,e^{-Nmathrm{tr}(-tfrac{h}{2}[A,B]^2+tfrac{1}{2}A^2+tfrac{g}{4}A^4+tfrac{1}{2}B^2+tfrac{g}{4}B^4)}$ at $g=h=1$, the bounds at cutoff $Lambda=11$ give $0.421783612le t_2le 0.421784687$ and $0.333341358le t_4le 0.333342131$, about six digits. The method also reproduces the known analytic answers at $h=0$ and $g=0$, maps a feasible region in the $(g,h)$ plane, and is extended to symmetry-breaking solutions by fixing the symmetry-breaking order parameter and scanning over it.

Load-bearing premise

The relaxation replaces the exact relation $X = x x^{T}$ by the weaker inequality $X succeq x x^{T}$, which enlarges the feasible set, so the multi-matrix precision claims rest on the authors' stated but unproven belief that this relaxation gap closes as the cutoff grows.

Editorial extensions

If this is right

  • Any single-trace moment of a large-$N$ multi-matrix integral can be bracketed by rigorous upper and lower bounds that improve monotonically with the word-length cutoff, with no statistical error.
  • The one-matrix quartic model is fully classified: a physical solution exists exactly for $gge -1/12$ at $mu=1$, and for $mu=-1$ the $Z_2$-breaking region $0<g<1/4$ has an explicitly known one-parameter family of solutions.
  • The phase boundary of the commutator-squared two-matrix model in the $(g,h)$ plane can be located numerically by bisection on feasibility, and the result is consistent with the known critical points at $h=0$ and $g=0$.
  • Spontaneous breaking of $Ato -A$, $Bto -B$ and $Aleftrightarrow B$ can be tested by maximising the corresponding order parameter; the authors find no evidence for it in the model with positive quadratic terms, but do recover the known breaking when the quadratic terms are flipped in sign.
  • The same loop-equation-plus-positivity scheme is proposed as an alternative to Monte Carlo for lattice Yang-Mills at large $N$, where it would give exact inequalities rather than statistical estimates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof for the one-matrix model suggests a natural conjecture the paper does not state: for any multi-matrix model, positivity of the full correlation matrix should exclude exactly those saddle points whose eigenvalue support leaves the appropriate real slice. If true, the bootstrap would be selecting the physical saddle by a general principle rather than by model-specific accident.
  • The observation that the relaxed two-matrix bounds coincide with the unrelaxed one-matrix bounds at $h=0$, within error bars at each cutoff, hints that mixed-word positivity conditions such as $langlemathrm{Tr},ABABrangle$ substitute for the information lost in relaxation. A quantitative version of that trade-off would let one predict how much cutoff is needed for a target precision.
  • Because the relaxation gap is the only uncontrolled step, the method's precision claims are empirically calibrated rather than proved. A systematic study of the gap on the analytically known slices ($g=0$, $h=0$, $h=infty$) as a function of cutoff would convert the six-digit claim into a certified error budget.
  • The authors note that their SDP is badly scaled and required double-double arithmetic; this suggests the practical ceiling on cutoff is set by conditioning rather than by problem size, so a principled rescaling of operators by their expected growth rates could buy several more digits at the same cost.
Original abstract

We propose the relaxation bootstrap method for the numerical solution of multi-matrix models in the large $N$ limit, developing and improving the recent proposal of H.Lin. It gives rigorous inequalities on the single trace moments of the matrices up to a given "cutoff" order (length) of the moments. The method combines usual loop equations on the moments and the positivity constraint on the correlation matrix of the moments. We have a rigorous proof of applicability of this method in the case of the one-matrix model where the condition of positivity of the saddle point solution appears to be equivalent to the presence of supports of the eigenvalue distribution only on the real axis and only with positive weight. We demonstrate the numerical efficiency of our method by solving the analytically "unsolvable" two-matrix model with $\mathrm{tr}[A,B]^2$ interaction and quartic potentials, even for solutions with spontaneously broken discrete symmetry. The region of values for computed moments allowed by inequalities quickly shrinks with the increase of the cutoff, allowing the precision of about 6 digits for generic values of couplings in the case of $\mathbb{Z}_2$ symmetric solutions. Our numerical data are checked against the known analytic results for particular values of parameters.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,729 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.61 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We propose the relaxation bootstrap method for the numerical solution of multi-matrix models in the large $N$ limit, developing and improving the recent proposal of H.Lin. It gives rigorous inequalities on the single trace moments of the matrices up to a given "cutoff" order (length) of the moments. The method combines usual loop equations on the moments and the positivity constraint on the correlation matrix of the moments. We have a rigorous proof of applicability of this method in the case of the one-matrix model where the condition of positivity of the saddle point solution appears to be equivalent to the presence of supports of the eigenvalue distribution only on the real axis and only with positive weight. We demonstrate the numerical efficiency of our method by solving the analytically "unsolvable" two-matrix model with $\mathrm{tr}[A,B]^2$ interaction and quartic potentials, even for solutions with spontaneously broken discrete symmetry. The region of values for computed moments allowed by inequalities quickly shrinks with the increase of the cutoff, allowing the precision of about 6 digits for generic values of couplings in the case of $\mathbb{Z}_2$ symmetric solutions. Our numerical data are checked against the known analytic results for particular values of parameters.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We propose the relaxation bootstrap method for the numerical solution of multi-matrix models in the large $N$ limit, developing and improving the recent proposal of H.Lin. contribution 0.98 0.91 0.20 asserted 1.00
s1 It gives rigorous inequalities on the single trace moments of the matrices up to a given "cutoff" order (length) of the moments. method 0.63 0.81 0.09 asserted 1.00
s2 The method combines usual loop equations on the moments and the positivity constraint on the correlation matrix of the moments. method 1.00 0.47 0.05 asserted 1.00
s3 We have a rigorous proof of applicability of this method in the case of the one-matrix model where the condition of positivity of the saddle point solution appears to be equivalent to the presence of supports of the eigenvalue distribution only on the real axis and only with positive weight. result 0.69 0.92 0.14 asserted 0.95
s4 We demonstrate the numerical efficiency of our method by solving the analytically "unsolvable" two-matrix model with $\mathrm{tr}[A,B]^2$ interaction and quartic potentials, even for solutions with spontaneously broken discrete symmetry. result 0.78 0.96 0.47 asserted 1.00
s5 The region of values for computed moments allowed by inequalities quickly shrinks with the increase of the cutoff, allowing the precision of about 6 digits for generic values of couplings in the case of $\mathbb{Z}_2$ symmetric solutions. result 1.00 0.95 0.13 asserted 0.99
s6 Our numerical data are checked against the known analytic results for particular values of parameters. method 0.49 0.90 0.06 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.94.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Coulomb gas interpretation for the eigenvalue configurations and a typical cut configuration. For a general solution of the SDP (2.15), we can have a complex cut at the maximum of the potential. The possible physical solutions correspond to filling n minima of such a potential with fractions N1, N2, . . . , Nn of these charges, such that Pn l=1 Nk = N. The eigenvalues then form a continuous distribution with n finite supports along the real axis. The SPE describes all extrema of the effective potential Veff = N X k V (xk) + log ∆2 (x1, . . . , zN ), (2.16) not only the minima but…
Figure 2. The allowed region for quartic model V (z) = 1 2 z 2 + g 4 z 4 for different cutoff Λ, compared with the analytic solution. Here we assuming W1 = 0, we also note that for g > 0, we didn’t plot Λ ≥ 5 since they are almost indistinguishable from the analytic solution on the figure. 10Here we assume that all the expectation values we study are real. This is actually a non-trivial result from the M → MT symmetry of the potential. Since this symmetry is always present for all the models considered in this article, we will implicitly assume this to be always true. 11Here we slightly
Figure 3. The comparison of numerical bootstrap with our analytic bootstrap results (the black point or the black curve on the figures) in Section 3.2 with µ = −1 in potential (3.6). They are representatives from different phases of the model, with g = 1, 1 10 , 1 30 (for the figures from above to below, respectively). We notice even visually that for g < 1 4 i.e. when the symmetry breaks, the exact solution is a non-convex set. the form Wij = Wi−j . It is called the Toeplitz matrix in linear algebra16. For this 16For the Hermitian integral the correlation matrix is of the form of the Hanke
Figure 4. The allowed region of t2 − t4 of model (5.1) with parameter g = 1, h = 1 for the cutoff Λ = 7, 8, 9, 10, 11. We recall the definition of Λ: the longest operators in the correlation matrix and in the loop equations have the length 2Λ. the upper and lower bound of t2 and t4. For Λ = 10: ( 0.421780275 ≤ t2 ≤ 0.421785491 0.333339083 ≤ t4 ≤ 0.333343006 (5.4) and for Λ = 11: ( 0.421783612 ≤ t2 ≤ 0.421784687 0.333341358 ≤ t4 ≤ 0.333342131 (5.5) We see here that for Λ = 11 we already have a six digits precision at g = h = 1. The Λ = 11 calculation is the largest problem in this work, it
Figure 5. Comparison with the exact analytic solution of model (5.1) with h = 0, i.e. two decoupled quartic one-matrix model. The lower plot is for Λ = 8. The comparison of our numerical results with analytic result is presented on
Figure 6. Comparison of the numerical bootstrap results with the exact analytic solution of the model (5.1) with g = 0. The lower plot is for Λ = 8. There is another fact which is not obvious from the
Figure 7. The numerical phase diagram of the model (5.1). The gray region is strictly excluded by our relaxation bootstrap method at Λ = 8. In the blue region the matrix potential is bounded from below and its boundary is located above the straight lines g = 0 and h = − 1 4 g. The red and green dots are the critical valus for h = 0 and g = 0. Some explanations for the plot
Figure 8. The allowed region for t2 when we fix g = 1 and vary h. In the region where the potential is bounded from below i.e. h ≥ −1 4 , we have a decent convergence whereas for h < − 1 4 the convergence gets much slower. 2. As already stated in the last paragraph, when we are not in the “bad region”, the minimization scheme converges much faster than the maximization scheme. So for a reasonable estimate of the operator expectation we should privilege the minimization scheme. 3. We also notice that for the region g > 0, h > 0, the convergence is excellent as expected, but there is a huge…
Figure 9. The allowed region for the fixed h = 1 and varying g. For the lower bound of the pink region Λ = 9, there are some numerical instabilities at −0.09 < g < 0. From careful inspection of our data at various values of Λ it seems that the lower bound at Λ = 9 should stabilize in this region at the value t2 ' 0.5 if no numerical instabilities happened in our SDP solver. anymore but there is rather a continuous family of solutions parametrized by t1. This is a non-convex set of exact solutions, so we don’t expect that our relaxation bootstrap method, as applied in the case of
Figure 10. The allowed t1 −t2 region for Λ = 6 and Λ = 9. The corresponding parameters in the model (5.24) are g = 1 100 , h = 0. The shaded region is the result of one-matrix bootstrap. The black line is the exact analytic solution described in Section 3.2. The green and red dots are the upper bounds and lower bounds of our relaxation method from scanning over t1. In
Figure 11. The allowed region for g = 1 30 and h = 1 15 . These numerical results are obtained for Λ = 8. 6 Conclusion and discussion In this work, we develop further the matrix bootstrap method pioneered in the papers [28, 30] and propose a crucial improvement – the relaxation procedure – applicable to a large class of multi-matrix problems and allowing to bootstrap them with a much higher precision. The relaxation transforms a Non-linear SDP, with the non-linearity due to the structure of loop equations, to the usual, linear SDP. We demonstrate the efficiency of our approach on the analyti
Figure 12. The left plot shows the sign of C(x) in the neighborhood of a positive cut. The right plot shows there must be at least one zero, or generally odd number of zeros, between two cuts to fix the sign. consequence is that we must have at least one zero, or generally odd number of zeros, between two positive cuts to fix the sign. This is also illustrated in
Figure 13. The only possible cut configuration for asymptotic (+, −) Here we split the discriminant into the part depending on W1 and W2: D1(x) = gx3 − x 2 − 4gx2 + 4g (B.3) and and the part depending only on g: D2(x) = 4g (W1x + W2) . (B.4) D2(x) is a straight line with negative intercept. As depicted in
Figure 14. The only possible cut configuration for µ = 1 and g < 0. The critical case is g = − 1 12 , below that, there is no more bootstrap solution. From the
Figure 15. 4 solutions under the condition 1 when g = −1/15. The blue line is the functionD1(x) which is independent of moments W1, W2, whereas other color straight lines are different solution of D2(x). We see only the green line has the correct configuration of zeros depicted in

Figures reproduced from arXiv:2108.04830 by the authors.

References
How the literature uses it

53 arXiv papers cite this in 110 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 56 sentences without a confident label

Builds on it 1 paper
  • This can be encoded in a positive semidefinite matrix: $([ 1 x; x y; ]) ≽0.$ In addition, we introduce some new positivity relations (beyond what was done in [ref] ) by imposing $y ≤(P_I P_I)(P_J P_J)$ since positivity requires $(P_I P_I)^2 ≥P_I P_I ^2$.

    High-Precision Bootstrap of Multimatrix Quantum Mechanics · Nonlinear relaxation · 0.85

Uses its method or tool 6 papers
Compares with it 7 papers
  • As we make use of the fully nonlinear loop equations without using the relaxation procedure, we are able to surpass the accuracy of the positive bootstrap results in [ref] .

    Analytic trajectory bootstrap for matrix models · Introduction · 1.00

  • We considered this model with $g=h=1$ using MC methods and show in Fig. [ref] that the results obtained are consistent with Ref. [ref] .

    Introduction to Monte Carlo for Matrix Models · Hoppe-type matrix models: Confirming bootstrap results · 1.00

  • In the planar limit, for $g>0$ ([ref]) evaluates to[Note that while a naive power counting might have suggested that the MQM is weakly coupled at high temperatures, the model at nonzero coupling $g$ has a different $→0$ asymptotic behavior from the Gaussian model at $g=0$. ] (see e.g. (3.7) of [ref] ) E() = N^2 [ 34 - - 23^2 + 23 ^12 (1+)^32 ] +O(^12), ≡48 g.

    Thermal Bootstrap of Matrix Quantum Mechanics · Comparison to analytic results · 0.90

Cites it as support 1 paper
Cites it as background 27 papers
  • Following the ideas developed in large $N$ matrix models [ref] [ref] , they have pursued a new bootstrap approach to problems in quantum mechanics [ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] .

    Bootstrapping microcanonical ensemble in classical system · Introduction · 0.93

  • Previously, this two-matrix model has been studied using the relaxation bootstrap [ref] and the Monte Carlo method [ref] .

    Analytic trajectory bootstrap for matrix models · Two-matrix model · 0.83

  • Recently, a novel bootstrap method for numerical calculations in matrix models [ref] and matrix quantum mechanics [ref] is proposed, and has been studied in the literature [ref] .

    Bootstrapping Calabi-Yau Quantum Mechanics · Introduction · 0.93

What it cites · 37 sentences without a confident label

Builds on 1 paper
  • This model interpolates between the solvable cases with $(A^2B^2)$ or $(BABA)$ interactions [ref] . * An obvious, and one of the most ambitious possible applications of our relaxation bootstrap methods is the lattice Yang-Mills theory.

    Two-Matrix model with ABAB interaction · Conclusion and discussion · 0.83

Uses the method or tool of 3 papers
Compares with 2 papers
Cites as background 14 papers
  • Then, in the large $N$ limit, the problem can be reduced to the saddle point calculation, significantly simplifying the problem of computation of that functional integral.[An (incomplete) review of such solvable matrix models can be found in [ref] . ] The basic example of such a simplification is the one matrix model [equation] solvable for any potential $V(x)$.

    Solvable Matrix Models · Introduction · 0.98

  • But this is not a good question since generally, apart from some solvable models where the loop equations close on a very limited subclass of operators (like in a two-matrix model [ref] or some \(n\)-matrix models [ref] ), the number of operators grows faster than the number of constraints, which means that the solution is a region in an extremely high dimensional space.

    Q-states Potts model on a random planar lattice · Physical constraints · 0.82

  • But this is not a good question since generally, apart from some solvable models where the loop equations close on a very limited subclass of operators (like in a two-matrix model [ref] or some \(n\)-matrix models [ref] ), the number of operators grows faster than the number of constraints, which means that the solution is a region in an extremely high dimensional space.

    Combinatorial Solution of the Two-Matrix Model · Physical constraints · 0.82

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