A determinant identity for the sum of contour integral matrices

Abstract: We derive an identity for the determinant of the sum of two $n\times n$ matrices, $A$ and $B$, whose entries are defined via contour integrals. Specifically, we consider $A(i,j)=\frac{1}{2π\mathrm{i}}\oint_0 z^{i-j-1}p_i(z)f_j(z)\mathrm{d} z$ and $B(i,j)= \frac{1}{2π\mathrm{i}}\int_Γ q_i(z)g_j(z) \mathrm{d} z$. Under suitable assumptions on the functions $p,q,f,g$, we show that $\det(A+B)$ can be expressed as a Fredholm determinant $\det(\mathrm{I} +K)$, where $K$ is an integral kernel acting on the contour $Γ$. This result generalizes a recent identity obtained in \cite{Baik-Liao-Liu26}.

• The paper establishes that the determinant of the sum of contour integral matrices can be expressed as a Fredholm determinant, providing a novel analytical tool.
• It rigorously proves the trace class property of the integral kernel under strict analyticity and integrability conditions on the involved functions.
• The results extend previous work and enable asymptotic and numerical analyses in stochastic models like TASEP, KPZ, and last passage percolation.

Determinant Identities for Sums of Contour Integral Matrices

Summary of Main Results

The paper "A determinant identity for the sum of contour integral matrices" (2604.24747) derives an exact identity for the determinant of the sum of two $n\times n$ matrices whose entries are defined via contour integrals. The main theorem establishes that under suitable analyticity and integrability assumptions on functions $p_i$, $q_i$, $f_j$, $g_j$, and a linking function $H$, the determinant $\det(A+B)$—with $A$ and $B$ composed of contour integrals—can be represented as a Fredholm determinant $\det(I+K)$, where $p_i$0 is an explicit integral kernel over the contour $p_i$1. This extends prior work involving determinant identities critical for the analysis of stochastic processes with integrability structures, specifically generalizing results from [Baik-Liao-Liu26].

Detailed Mathematical Constructs

The matrices under consideration are given by

• $p_i$2
• $p_i$3

with:

• $p_i$4, $p_i$5 analytic near the origin and normalized at $p_i$6;
• $p_i$7, $p_i$8 square integrable on $p_i$9;
• $q_i$0 analytic at $q_i$1 satisfies $q_i$2.

The primary identity proven is:

$q_i$3

where $q_i$4 has the form

$q_i$5

with nested contours and $q_i$6 ($q_i$7).

The paper gives rigorous proofs that $q_i$8 is a trace class operator, ensuring well-definedness of the Fredholm determinant, and demonstrates simplifications in the kernel for special cases (e.g., $q_i$9).

Numerical, Structural, and Analytical Highlights

• The Fredholm determinant representation is proven for highly general matrix forms, beyond just Toeplitz or Hankel structures.
• When $f_j$0, the kernel simplifies to a single integral involving a sum; this matches known results in particular cases, validating the generalization.
• Infinitely many functions $f_j$1 fulfill the necessary analytic and integral relations, which is essential for applications where $f_j$2 (and the resulting $f_j$3) encode stochastic initial conditions or combinatorial weights.

The paper emphasizes that the identity does not match the Borodin-Okounkov Fredholm determinant formula for Toeplitz determinants [Borodin-Okounkov00], with the current results covering fundamentally different algebraic and analytic scenarios.

Implications and Applications

The identity has direct implications in integrable probability and mathematical physics. In particular, it has been motivated by problems related to geodesic distributions in last passage percolation (LPP) and multipoint distributions in the KPZ fixed point with general initial conditions. The analytic structure allows for asymptotic analysis and exact computation of distribution functions in models such as TASEP, where previously only specific initial conditions were tractable. The kernel $f_j$4, especially its simplifications, provides a bridge between matrix determinant computations and operator-theoretic approaches that are standard in contemporary integrability.

Practically, this result yields a more flexible toolkit for expressing and analyzing determinants arising in stochastic models via Fredholm determinants, which are amenable to powerful techniques like Riemann-Hilbert analysis and steepest descent, as well as numerical evaluation.

Theoretical Significance and Future Directions

The work opens pathways for further generalizations:

• Extension to more complex matrix structures, such as block matrices of contour integrals;
• Adaptation to higher-dimensional contour integrals and analytic kernels, relevant in multivariate stochastic models;
• Investigation of connections to other integral operator identities in random matrix theory, determinantal point processes, and LPP.

Furthermore, the result sharpens existing understanding of determinants in the context of analytic combinatorics and operator theory, highlighting the interplay between matrix analysis and functional analytic techniques. The existence of infinitely many suitable $f_j$5 functions suggests that this determinant identity can be adapted to a wide range of models by appropriate choice, potentially leading to exact asymptotic results in new models within KPZ universality.

Conclusion

The paper rigorously derives a determinant identity for sums of contour integral matrices, establishing equivalence with a Fredholm determinant structured by an explicitly constructed kernel. This generalization supports both theoretical and practical advances in integrable probability, expanding the analytic framework for matrix determinants and connecting it with operator-theoretic representations crucial for asymptotic analysis and exact computations. The identity, its generality, and the explicit kernel construction constitute substantive progress for researchers leveraging determinant methods in stochastic modeling and combinatorial probability (2604.24747).