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arxiv: 2605.01202 · v1 · submitted 2026-05-02 · 🧮 math.NA · cs.NA

Sampling Pfaffian point processes and the symplectic Arnoldi method

Alan Edelman , Sungwoo Jeong , Simeon Schaub This is my paper

Pith reviewed 2026-05-09 18:57 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Pfaffian point processesskew-symmetric Choleskysymplectic Arnoldibeta ensemblesrandom matrix samplingorthogonal ensemblessymplectic ensemblesexact sampling
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The pith

A skew-symmetric analogue of the Cholesky factorization gives an exact sampling algorithm for Pfaffian point processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an exact sampling procedure for Pfaffian point processes whose correlation functions are Pfaffians of a skew-symmetric kernel matrix. The procedure rests on performing a skew-symmetric version of the Cholesky factorization on that kernel, then drawing the points from the resulting triangular factors. A symplectic Arnoldi iteration is supplied to construct the 2x2 matrix kernels needed for beta=1 and beta=4 polynomial ensembles when only a weight function is given. A sympathetic reader would care because the method turns previously indirect or approximate generation of points from orthogonal and symplectic ensembles into a direct, exact computation.

Core claim

The authors establish that any Pfaffian point process can be sampled exactly once its skew-symmetric kernel matrix is available, by computing a factorization K = L J L^T in which L is lower triangular and J is a fixed block-diagonal matrix of 2-by-2 symplectic blocks; the diagonal entries of L and the off-diagonal signs supplied by J then determine the locations of the random points. They further show that the required kernels for polynomial ensembles are obtained by running a symplectic Arnoldi process that produces skew-orthogonal polynomials with respect to the given weight.

What carries the argument

The skew-symmetric Cholesky factorization, which decomposes the kernel matrix into a lower-triangular factor and a fixed block-diagonal sign matrix so that the random points can be read off from the triangular entries.

If this is right

  • Eigenvalues of the Gaussian orthogonal and symplectic ensembles can be sampled exactly for any finite N.
  • Points from the symmetric corner growth model become directly generatable.
  • Finite-N beta=1 and beta=4 Airy point processes and the associated Tracy-Widom laws can be sampled.
  • Kernels for general-weight polynomial ensembles are constructed via the symplectic Arnoldi iteration without explicit orthogonalization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same factorization may allow exact sampling of other determinantal or Pfaffian structures that appear in combinatorial probability.
  • Large-scale Monte Carlo studies of integrable models in random matrix theory become feasible once the factorization cost is controlled.
  • Extension to non-polynomial ensembles would require only a stable kernel evaluation routine.

Load-bearing premise

The skew-symmetric Cholesky factorization exists, remains numerically stable, and can be computed without breakdown for the kernels that arise in the target ensembles.

What would settle it

Apply the algorithm to the 2-point Pfaffian process obtained from the 2-by-2 Gaussian orthogonal ensemble and verify whether the histogram of sampled eigenvalues matches the known exact distribution to machine precision.

Figures

Figures reproduced from arXiv: 2605.01202 by Alan Edelman, Simeon Schaub, Sungwoo Jeong.

Figure 1
Figure 1. Figure 1: Comparing different methods for computing skew-orthogonal polyno￾mial bases, it is evident that our symplectic Arnoldi method signifi￾cantly outperforms the method based on the skew-symmetric Cholesky factorization (in black) in terms of numerical accuracy. Classical sym￾plectic Gram-Schmidt without reorthogonalization (CSGS) also failed to maintain skew-orthogonality for larger n, while classical symplec￾… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of different normalizations for skew-orthogonalization. In all cases, classical symplectic Gram-Schmidt with iterated reorthogo￾nalization (CSGS IR) and η = 0.75 was used n 5 10 20 50 M a x ortho g onality error 10−10 100 1010 Numerical Accuracy n 5 10 20 50 Time (s) 10−4 10−3 10−2 Computation Time Scaling of our Arnoldi method with different ESR using ⟨ ⋅ , ⋅ ⟩w 1 , w(x) = 0.5x ESR1 ESR2 ESR3m … view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of different normalizations for skew-orthogonalization on an exponentially decaying weight function. In all cases, classical sym￾plectic Gram-Schmidt with iterated reorthogonalization (CSGS IR) and η = 0.75 was used MSGS. The difference between reorthogonalizing once at every iteration (CSGS2) and iterated reorthogonalization (CSGS IR) were almost negligible, with CSGS2 perhaps having a slight e… view at source ↗
Figure 4
Figure 4. Figure 4: Shown above are the skew-orthogonal polynomials with respect to the discrete weight w(x) = 0.8 x/2 , x ∈ Z≥0, corresponding to symmetric corner growth with p = 0.8. Below is the weight function itself. Let A = view at source ↗
Figure 5
Figure 5. Figure 5: The distribution of F(10), the time it takes for symmetric corner growth to reach the square at (10, 10), alongside the same distribution sampled from the corresponding PfPP. Johannson derives its distribution as P[F(N) ≤ t] = 1 Z (1) N X h∈N N max{hj }≤t+N−1 Y 1≤i<j≤N |hj − hi | Y N i=1 q hi/2 . This is exactly of the form of the discrete Coulomb gas for β = 1, so we can construct K from skew-orthogonal p… view at source ↗
Figure 6
Figure 6. Figure 6: The first six skew-orthogonal polynomials with respect to the GOE skew-inner product. These skew-orthogonal polynomials have already been derived analytically, see [Meh04], but we recompute them numerically in order to demonstrate our symplectic Arnoldi method on continuous kernels. Since adaptive quadrature methods struggle with the discontinuity at x = y, the inner integral is split in two and both trian… view at source ↗
Figure 7
Figure 7. Figure 7: Shown above is the distribution of the maximum eigenvalues of the N = 10 GOE, normalized as described in (14) to match the β = 1 Tracy-Widom distribution as N → ∞. The solid blue lines and error bars correspond to the values sampled from the discretized PfPP using our modified skew-Cholesky method. The dashed red line corresponds to a histogram of the maximum eigenvalues directly sampled from the GOE matri… view at source ↗
Figure 8
Figure 8. Figure 8: The first six skew-orthogonal polynomials with respect to the GSE skew-inner product (13) view at source ↗
Figure 9
Figure 9. Figure 9: The above histograms were again produced as already described for view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of different continuous sampling algorithms from 1000 samples taken for each of the methods. All methods, except for MALA-within-Gibbs, show similarly accurate results, which strug￾gles due to strong autocorrelation between samples due to the nature of the Metropolis-adjusted Langevin algorithm. This means MALA￾within-Gibbs requires a larger number of samples to give accurate density estimates.… view at source ↗
Figure 11
Figure 11. Figure 11: The density of the second largest eigenvalue of the Airy1 point pro￾cess. The histogram is obtained from Algorithm 1, and the red line is obtained from the formula for f (1) λ2 (s) described in (15) view at source ↗
Figure 12
Figure 12. Figure 12: The density of the second largest eigenvalue of the Airy4 point pro￾cess. The histogram is obtained from Algorithm 1, and the red line is obtained from the formula for f (4) λ2 (s), which is defined in the same as f (1) λ2 (s) in (15). References [AFNVM00] M Adler, PJ Forrester, T van Nagao, and P Van Moerbeke. Classical skew orthogonal polynomials and random matrices. Journal of Statistical Physics, 99(1… view at source ↗
read the original abstract

We present an exact sampling algorithm for Pfaffian point processes based on a skew-symmetric analogue of the Cholesky factorization. This algorithm enables efficient sampling of a wide range of statistics arising in random matrix theory and combinatorics. For instance, we can sample eigenvalues of the orthogonal and symplectic ensembles ($\beta = 1,4$). In addition, we introduce a symplectic Arnoldi method for computing skew-orthogonal polynomials associated with a general weight function. This method can be used to efficiently construct the $2 \times 2$ matrix valued skew-symmetric kernels that arise in $\beta = 1,4$ polynomial ensembles. We illustrate our approach with several numerical examples and experiments, including the symmetric corner growth model, the finite-$N$ Gaussian (Hermite) orthogonal and symplectic ensembles, and the $\beta = 1,4$ Airy point processes and Tracy-Widom distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents an exact sampling algorithm for Pfaffian point processes based on a skew-symmetric analogue of the Cholesky factorization. It also introduces a symplectic Arnoldi method for computing skew-orthogonal polynomials to construct the 2×2 skew-symmetric kernels arising in β=1,4 polynomial ensembles. The approach is illustrated with numerical examples for the symmetric corner growth model, finite-N Gaussian orthogonal and symplectic ensembles, and β=1,4 Airy point processes with associated Tracy-Widom distributions.

Significance. If the exactness and numerical stability claims hold, the work would provide an efficient, non-approximate sampler for a class of point processes central to random matrix theory and integrable combinatorics, enabling direct Monte Carlo studies of eigenvalue statistics and growth models that are currently limited by the lack of exact samplers.

major comments (2)
  1. [Algorithm section (skew-symmetric Cholesky construction)] The exactness of the sampler rests on the skew-symmetric Cholesky factorization existing without breakdown (zero pivots) and without pivoting for all kernels in the claimed applications, including the Airy kernel at β=1,4 and finite-N GOE/GSE kernels. The manuscript must supply either a proof of existence/stability for these kernels or explicit numerical verification that no pivoting is required and that the Pfaffian structure is preserved; absent this, the central 'exact sampling' claim cannot be verified for the target ensembles.
  2. [Symplectic Arnoldi method section] The symplectic Arnoldi method is presented as a tool to build the kernels, but no convergence analysis, backward stability bounds, or comparison to existing methods for skew-orthogonal polynomials is given. This is load-bearing because the kernels must be constructed to sufficient accuracy for the subsequent factorization to remain exact.
minor comments (2)
  1. [Numerical experiments] Numerical illustrations should report quantitative diagnostics (e.g., Kolmogorov-Smirnov distances to known Tracy-Widom laws or exact marginals) rather than visual overlays alone.
  2. [Introduction / Preliminaries] Notation for the 2×2 block skew-symmetric kernel K should be introduced with an explicit definition of the Pfaffian and the associated point process measure early in the paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the insightful comments and for highlighting the importance of verifying the exactness and stability of our proposed algorithms. We respond to each major comment below and outline the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: The exactness of the sampler rests on the skew-symmetric Cholesky factorization existing without breakdown (zero pivots) and without pivoting for all kernels in the claimed applications, including the Airy kernel at β=1,4 and finite-N GOE/GSE kernels. The manuscript must supply either a proof of existence/stability for these kernels or explicit numerical verification that no pivoting is required and that the Pfaffian structure is preserved; absent this, the central 'exact sampling' claim cannot be verified for the target ensembles.

    Authors: We agree that this is a critical point for substantiating the exact sampling claim. Although the manuscript demonstrates successful sampling for the target ensembles (symmetric corner growth, finite-N GOE/GSE, and β=1,4 Airy processes) without reported issues, we did not include explicit checks for pivot non-zero status or Pfaffian preservation. In the revised version, we will add numerical verification in the examples section, reporting the minimum pivot sizes encountered (which are positive) and confirming that the sampled statistics align with known results, thereby preserving the structure. A general theoretical proof for all possible kernels is not feasible within the scope, but for the specific applications, this will suffice. revision: yes

  2. Referee: The symplectic Arnoldi method is presented as a tool to build the kernels, but no convergence analysis, backward stability bounds, or comparison to existing methods for skew-orthogonal polynomials is given. This is load-bearing because the kernels must be constructed to sufficient accuracy for the subsequent factorization to remain exact.

    Authors: The symplectic Arnoldi method serves as a practical means to compute the required skew-orthogonal polynomials for kernel construction. Its utility is shown through the numerical examples where the resulting samples reproduce expected distributions accurately. We recognize the absence of formal analysis. We will revise the manuscript to include a short subsection discussing the observed numerical convergence in our experiments, citing relevant literature on skew-orthogonal polynomials, and clarifying that while full backward stability bounds are not derived here, the method's accuracy is validated by the downstream sampling results. This addresses the concern without shifting the paper's focus. revision: partial

Circularity Check

0 steps flagged

No circularity: algorithmic construction is self-contained

full rationale

The paper presents constructive algorithms—an exact sampler via skew-symmetric Cholesky factorization of the 2x2 block kernel and a symplectic Arnoldi method for skew-orthogonal polynomials—rather than any derivation chain that reduces a claimed result to its own fitted inputs or self-citations. No self-definitional steps, renamed empirical patterns, or load-bearing uniqueness theorems appear; the methods are defined directly in terms of standard linear-algebra operations on the given kernels, and their exactness for the listed ensembles (GOE/GSE, Airy, corner growth) is asserted as a property of the construction that can be verified independently by implementation and numerical test. The central claims therefore remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, no free parameters, ad-hoc axioms, or invented entities are introduced; the work relies on standard linear-algebraic factorizations whose existence is assumed for the relevant kernels.

pith-pipeline@v0.9.0 · 5451 in / 1120 out tokens · 26592 ms · 2026-05-09T18:57:34.788830+00:00 · methodology

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Reference graph

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