Pal's permanent conjecture: proof for block uniform matrices

Abstract

Consider a symmetric function 𝒞(x,y) on [0,1]×[0,1] which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let A⁽ⁿ⁾ = (a⁽ⁿ⁾_i,j : i,j ∈ [n]) be the matrix with entries a⁽ⁿ⁾_i,j = exp(−𝒞(i/n,j/n)). Soumik Pal conjectured the asymptotics perm(A⁽ⁿ⁾)/n!∼ exp(n Λ[𝒞])/ √(𝒟[𝒞]) as n → ∞ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional Λ[𝒞] is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is ∫₀¹ ∫₀¹ (α(x)+β(y)) dx dy where α(x)+β(y) is chosen such that ρ(x,y) := exp(−𝒞(x,y)−α(x)−β(y)) has uniform marginals. The algebraic term 𝒟[c] is given by Peter McCullagh's formula for doubly stochastic matrices: det_F(I+J−T^*T), the Fredholm determinant, where I is the identity on L²([0,1]), Jf(x) ≡ ∫₀¹ f(z) dz (for all x) and Tf(x) = ∫₀¹ ρ(x,y) f(y) dy. We prove the conjecture for functions 𝒞 that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.

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BibTeX RIS

BibTeX

@misc{ottolini2026pals,
  title = {{Pal's permanent conjecture: proof for block uniform matrices}},
  author = {Andrea Ottolini and Shannon Starr},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.25274},
  eprint = {2605.25274},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.25274},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Pal's permanent conjecture: proof for block uniform matrices
AU  - Andrea Ottolini
AU  - Shannon Starr
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2605.25274
UR  - https://arxiv.org/abs/2605.25274
AB  - Consider a symmetric function 𝒞(x,y) on [0,1]×[0,1] which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let A^(n) = (a^(n)_i,j : i,j ∈ [n]) be the matrix with entries a^(n)_i,j = exp(−𝒞(i/n,j/n)). Soumik Pal conjectured the asymptotics perm(A^(n))/n!∼ exp(n Λ[𝒞])/ √(𝒟[𝒞]) as n → ∞ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional Λ[𝒞] is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is ∫_0^1 ∫_0^1 (α(x)+β(y)) dx dy where α(x)+β(y) is chosen such that ρ(x,y) := exp(−𝒞(x,y)−α(x)−β(y)) has uniform marginals. The algebraic term 𝒟[c] is given by Peter McCullagh's formula for doubly stochastic matrices: det_F(I+J−T^*T), the Fredholm determinant, where I is the identity on L^2([0,1]), Jf(x) ≡ ∫_0^1 f(z) dz (for all x) and Tf(x) = ∫_0^1 ρ(x,y) f(y) dy. We prove the conjecture for functions 𝒞 that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.
ER  -

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