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Multideterminantal measures

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Pith's one-line read Pure k-determinantal measures are exactly sign-restricted Grassmannian tuples.

desk verdict New framework with a false central example and a characterization that is only half-proved; worthwhile to fix, but not citable as is. read the letter →

arxiv 2501.18349 v2 pith:XG36BPAK submitted 2025-01-30 math.PR

Richard Kenyon This is my paper · ORCID
classification math.PR MSC 60G5560C0515A1514M1505C70
keywords multideterminantalmeasuresdeterminantalGrassmannianPlückercoordinatesPfaffiangraphsrandompermutationsdimermodelVinnikovvariety
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces k-determinantal measures, probability measures on colorings of n sites with k colors, whose point probabilities are determinants of matrices assembled from k matrices summing to the identity. For k=2 these are classical determinantal measures. The central result is a structural classification: a pure k-determinantal measure (one whose matrices have ranks summing to n) is determined by a k-tuple of Grassmannian elements, and the measure is a genuine probability measure exactly when certain products of maximal minors have the right signs. A special case constructs every pure 2-determinantal measure from two Grassmannian elements with matching Plücker signs. The paper also classifies determinantal probability measures on the symmetric group as exactly the measures coming from bipartite Pfaffian graphs with positive edge weights.

What carries the argument

The load-bearing mechanism is the factorization of point probabilities into products of minors of the inverse matrix L. Given projections A_i with images V_i, one forms M whose column blocks span V_i, sets L=$M^{{-1}}$, and reads each color sequence π as a placement of the index blocks U_i. The identity Pr(π)=(-1)^π (det M) ∏_i L_{π(U_i)}^{U_i} converts the determinant definition (2) into pure Grassmannian data. The companion Pfaffian mechanism for permutations uses rank-one decompositions A_i=u_i v_i^t and the Kasteleyn matrix V, whose nonnegative expansion terms force V to be the Kasteleyn matrix of a bipartite Pfaffian graph.

What would settle it

Take a pure 3-determinantal measure with block sizes (3,3,3) and random rank-one projection matrices A_1,A_2,A_3 summing to the identity; compute Pr(π) by the determinant definition (2) and by the product-of-minors formula (8) for a permutation π. Any discrepancy disproves the factorization and with it the Grassmannian characterization.

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Extended reading notes

Core claim

The paper claims that pure k-determinantal measures admit a complete description in terms of Grassmannian geometry. With block sizes n_1,...,n_k summing to n, the point probabilities factor as products of maximal minors of a matrix L (the inverse of the matrix whose column blocks span the image subspaces), up to a global sign and constant: Pr(π)=(-1)^π Q ∏ L_{π(U_i)}^{U_i}. Therefore the kernel of a pure k-determinantal measure is exactly a k-tuple of Grassmannian elements L_i ∈ Gr_{n_i,n} whose maximal minors, taken in these products, satisfy the sign restrictions that make all probabilities nonnegative. For k=2 this reduces to two elements of Gr_{n_1,n} with Plücker coordinates of the same sign. Separately, for k=n with rank-one matrices, the measures are supported on permutations, and the paper proves they are precisely those obtained from a Kasteleyn matrix of a bipartite Pfaffian graph with positive edge weights.

Load-bearing premise

The proof of Theorem 7 assumes a general determinant factorization—point probabilities equal a signed constant times a product of minors of L—for all block sizes n_1,...,n_k, but the paper only demonstrates it for the case (2,2,2).

Editorial extensions

If this is right

  • Every pure k-determinantal measure is encoded by a k-tuple of Grassmannian elements, so the classification reduces to Plücker sign patterns rather than arbitrary matrix kernels.
  • For k≥3, the product formula implies a support restriction: among the six colorings obtained by permuting three distinct colors on three sites, only one cyclic triple can have positive probability—the other three cannot all be positive.
  • All pure 2-determinantal measures can be constructed from a pair of same-sign Grassmannian elements, giving a full albeit non-parameterized description; Mnëv universality suggests no simple parameterization exists.
  • Determinantal random permutations are exactly those arising from bipartite Pfaffian graphs with positive edge weights, so the Heawood graph yields a uniform measure on 24 permutations.
  • Symmetric k-determinantal measures with commuting matrices behave like sums of independent biased k-sided dice, with the characteristic polynomial factorizing into linear terms.

Reading between the lines

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

  • If the determinant factorization in Theorem 7 is checked for block sizes beyond (2,2,2), the Grassmannian characterization and the support restriction would be placed on solid ground; a counterexample would localize exactly where the classification breaks.
  • The sign-restriction condition on Plücker coordinates is reminiscent of total positivity, so pure k-determinantal measures may fit into the cluster-algebra structure of the positive Grassmannian, potentially yielding explicit coordinate systems for DET_{n,kn}.
  • The Vinnikov-curve property for k=3 might connect the open problem of which polynomials are characteristic polynomials to the theory of hyperbolic polynomials and real stable matrices, giving a necessary condition sharp enough to test.
  • For k=n, the Pfaffian classification suggests that determinantal permutations form a 'Pfaffian' subclass of the much larger set of permutation processes, possibly overlapping with existing models such as the Mallows distribution only at special parameter values.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces 'k-determinantal measures' on [k]^n, whose point probabilities are determinants of matrices whose j-th column is taken from one of k matrices A_1,...,A_k summing to the identity, and studies the case where the ranks sum to n ('pure' measures). It proposes a description of pure k-determinantal measures by sign conditions on Plücker coordinates of k Grassmannian elements, gives constructions from the positive Grassmannian, the bipartite dimer model, and the spanning tree model, and classifies rank-one (permutation) k-determinantal measures on S_n as measures coming from bipartite Pfaffian graphs. The main new tools are a product-of-minors formula for point probabilities (Theorem 7) and a sign-cancellation argument for k=2 (Section 6.2).

Significance. If established, the paper would provide a broad multideterminantal generalization of determinantal processes, with genuine connections to total positivity, dimers, and Pfaffian graphs; the characterization of pure measures through Grassmannian data would be a useful structural result, and Theorem 6 would give a complete description of determinantal permutation measures. The paper also formulates interesting open problems. The k=2 construction and the Pfaffian classification are the most convincing parts. However, in the current version a central example is false and the main factorization/converse is not proved, so the significance cannot be assessed on the basis of the text as written.

major comments (4)
  1. [Section 3.3, Theorem 5] Theorem 5 is false as stated: the symmetric normalization A_i=(Δ')^{-1/2}Δ'_i(Δ')^{-1/2} does not give the spanning-tree coloring measure. For example, take a graph with root 0 and non-root vertices 1,2, with colored conductances: edges 1--0 have (red,blue)=(1,1), edge 2--0 has (2,0), and edge 1--2 has (0,2). The tree-coloring probabilities are 1/6, 1/2, 1/6, 1/6 for (RR), (BR), (RB), (BB), respectively, whereas the matrices A_i defined in Theorem 5 give 1/6, 1/3+1/(4√3), 1/3−1/(4√3), 1/6. The proof applies the Directed Matrix Tree Theorem to det(Δ^1_{σ(1)},...,Δ^{n-1}_{σ(n-1)})/det Δ, which is the measure with A_i=Δ^{-1}Δ_i, not the symmetrized matrices. This invalidates the claimed spanning-tree example, Figure 1, and the use of the spanning-tree model as a symmetric k-determinantal measure in Section 4.
  2. [Section 6, Theorem 7] Theorem 7 is not proved. The proof verifies equation (9) for the single block-size vector (n1,n2,n3)=(2,2,2) and a single word π, then states that 'the general pattern can be seen'. This is a determinant factorization that must be proved for arbitrary block sizes n_i and arbitrary π. Because Theorem 7 is the basis for Corollary 8, for the support restriction in Section 6.1, and for the k=2 construction in Section 6.2, an unsupported example is not an adequate proof of a load-bearing statement. The identity may be true, but as written it is a conjecture backed by one instance.
  3. [Section 6, Corollary 8 and abstract] The claimed characterization is only one-way for k≥3. Corollary 8 states that every pure k-determinantal measure can be constructed from Grassmannian elements L_i whose products in (8) are nonnegative, but no converse is stated or proved: one is not told that arbitrary L_i satisfying the sign condition can be completed to an invertible n×n matrix L whose row blocks are the L_i, nor that (7) then defines a pure k-determinantal measure. The missing direct-sum hypothesis is necessary: for k=3, n1=n2=n3=1, the three lines L_i=span(e1) in R^3 make every product in (8) zero, hence nonnegative, but they cannot be completed to an invertible L and do not define a pure 3-determinantal measure. Thus the abstract's phrase 'characterize kernels' is not justified by the theorems proved in the paper.
  4. [Section 4, first paragraph] The assertion that B_i+λI form symmetric k-determinantal measures is not justified. The normalization of an unnormalized k-determinantal measure in Section 2.2 is left multiplication by (Σ A_i)^{-1}; this does not preserve symmetry unless that matrix commutes with every B_i. Since Theorem 5 is false, the symmetric spanning-tree example and the Vinnikov-curve discussion in Section 4.1 and Figures 4--5 rest on an invalid example. A corrected treatment must specify whether Section 4 concerns unnormalized or normalized measures and must provide a valid symmetric example if the Vinnikov-curve claims are to be retained.
minor comments (4)
  1. [Section 6, Theorem 7 notation] The notation π(U_j) is ambiguous: the U_j were earlier defined as row blocks of [n], but π(U_j) in the theorem denotes the preimage of color j, i.e., the positions where π takes value j. A distinct symbol, for example P_j={i:π(i)=j}, would avoid the conflict.
  2. [Section 6, Corollary 8] Corollary 8 should be labeled as a necessary condition; the phrase 'can be constructed from' overstates the logical direction that is actually proved.
  3. [Section 4.1, Figures 4 and 5] The Vinnikov-curve figures depend on Theorem 5; if Theorem 5 is corrected or replaced, these figures must be regenerated from a valid symmetric k-determinantal measure or the captions must be changed.
  4. [Section 2.2, equation (5)] The statement 'det M > 0' in the normalization step needs a short justification: since the determinants in (2) are nonnegative and not all zero, one needs an argument that det M has a well-defined sign; otherwise the normalization must be phrased in terms of |det M| or a choice of orientation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivations are self-contained and rest on external theorems, with no fitted values or self-referential definitions.

full rationale

The paper defines multideterminantal measures by determinantal formulas (equation 2) and then derives properties from those definitions using multilinearity, the directed matrix tree theorem, and prior dimer and Pfaffian results. The central structural result, Theorem 7, is an asserted determinant factorization; even if its proof is only sketched by example and may be incomplete, this is a correctness or completeness concern, not a circular one, because the formula is not assumed as an input. The pure-case characterization in Section 6.2 is proved by constructing L from Grassmannian elements with same-sign Plücker coordinates and verifying positivity using Lemmas 9 and 10; it does not define the target measures as those constructed from such elements. The permutation classification (Theorem 6) cites Vazirani-Yannakakis for the Pfaffian characterization, which is an external result, and Kenyon's earlier dimer result [6] is prior independent work with stated assumptions, not the present paper's claim. No parameter is fitted to data and no prediction is renamed from a fit. The apparent gap flagged in review, namely that Corollary 8 is stated as a characterization though only the forward direction plus the k=2 converse are proved, concerns possible non-equivalence of the stated conditions for k>=3, but it does not make the derivation circular. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces new definitions (k-determinantal measures, pure measures, Vinnikov varieties) but no fitted parameters or new physical entities. It depends on standard external theorems and one unproven determinant factorization (Theorem 7).

assumptions (6)
  • standard math Directed Matrix Tree Theorem (Chaiken [4])
    Used in Section 3.3 proof of Theorem 5 to identify column determinants with weighted sums of colored spanning trees. The theorem itself is not in question, but its application with the symmetric normalization is incorrect.
  • standard math Vazirani-Yannakakis theorem [16]: a matrix whose nonzero determinant expansion terms all have the same sign is a Kasteleyn matrix of a bipartite Pfaffian graph
    Used in Section 5 to classify determinantal permutation measures.
  • standard math Kasteleyn theory for dimer counts [5,7]
    Used in Section 3.2 to assert K is the kernel of the dimer inclusion measure.
  • domain assumption Generic position assumption for pure 2-determinantal construction: all Plücker coordinates nonzero, nongeneric by limiting
    Section 6.2. Standard but unproved continuity of the measure construction under limits when Plücker coordinates vanish.
  • standard math Mnev universality theorem [11]
    Used in Section 6.2 to argue no simple parameterization of sign patterns exists.
  • ad hoc to paper The determinant factorization in Theorem 7 (product of minors of L with sign (-1)^pi)
    This is the key unproven lemma; the paper only demonstrates a single (2,2,2) example and states the general pattern. The entire pure-case characterization and support restriction depend on it.

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Pith. "Pith review of Multideterminantal measures." pith.science (2026). https://pith.science/paper/XG36BPAK

@misc{pith2026250118349,
  author       = {Pith},
  title        = {Pith review of: Multideterminantal measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XG36BPAK}},
  note         = {Machine review of arXiv:2501.18349}
}
abstract

We define multideterminantal probability measures, a family of probability measures on $[k]^n$ where $[k]=\{1,2,\dots,k\}$, generalizing determinantal measures (which correspond to the case $k=2$). We give examples coming from the positive Grassmannian, from the dimer model and from the spanning tree model. We characterize kernels of \emph{pure} $k$-determinantal measures as those arising from $k$-tuples of Grassmannian elements whose maximal minors have certain sign restrictions. As a special case we construct all kernels of pure determinantal measures via a pair of elements of $Gr_{n_1,n}$ having corresponding Pl\"ucker coordinates of the same signs. We also define and completely characterize determinantal probability measures on the permutation group $S_n$.

Figures

Figures reproduced from arXiv: 2501.18349 by the authors.

Figure 1
Figure 1. A 3-determinantal measure (with 3 colors, red, blue and green) on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A small bipartite graph with Kasteleyn signs (edges without labels [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. A spanning tree with wired boundary and nonuniform conduc [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: A Vinnikov curve of degree n = 7 arising from the spanning tree 3-determinantal measure on K8 with randomly chosen conductances. -10 -5 0 5 10 -10 -5 0 5 10 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: The Vinnikov curve from a 3-determinantal measure on [3] [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The Heawood graph. Let ni be the rank of Ai . Let M be an n × n matrix whose first n1 columns span V1, next n2 columns span V2, and so on. This matrix M is well-defined up to the right action by GLn1 (R) × · · · × GLnk (R), which performs column operations on the first…

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