Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · hep-th/9306042

Fredholm Determinants, Differential Equations and Matrix Models

Craig A. Tracy·Harold Widom
View original source ↗Read source document ↗Export metadata ↓

Abstract

Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices.

Explore related subjects

hep-thmath-phmath.MPnlin.SIsolv-int

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Craig A. Tracy, Harold Widom. 1999-02-02. Fredholm Determinants, Differential Equations and Matrix Models. https://doi.org/10.1007/bf02101734

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Holographic Schwinger effect with Translational Symmetry Breaking

We investigate the holographic Schwinger effect in a background with translational symmetry breaking (TSB) at finite chemical potential. The gravitational background is characterized by two independent parameters: the TSB parameter \(α\), which controls momentum relaxation, and the chemical potential \(μ\), which determines the finite density of the dual field theory. Using the potential analysis method, we derive the total potential governing the pair production process and examine its dependence on \(α\), \(μ\), the external magnetic field, and the ratio \(β=E/E_c\). Our results show that the effects of \(α\) and \(μ\) on the Schwinger process strongly depend on the dynamical regime. In the subcritical regime, increasing either \(α\) or \(μ\) lowers the potential barrier and facilitates pair production. However, near and above the critical electric field, the roles of these two parameters become qualitatively different. While increasing the chemical potential lowers the total potential and enhances the Schwinger pair production process, increasing the translational symmetry breaking parameter shifts the potential upward and suppresses the production process. We further show that, at fixed $β=E/E_c$, a perpendicular external magnetic field lowers the effective potential barrier and thereby facilitates the Schwinger process, while the physical electric field changes accordingly through the magnetic-field dependence of $E_c$. The corresponding pair production rate is not independently calculated; instead, its qualitative behavior is characterized through a proxy derived from the total potential. Overall, our analysis provides a comprehensive picture of how translational symmetry breaking, finite density, and external magnetic fields influence holographic non-perturbative pair production.

hep-th↗
arXiv2026

One-loop Corrected Holographic Shear Viscosity to Entropy Density Ratio at Low Temperatures

Near-extremal black holes contain infrared-enhanced quantum fluctuations in their near-horizon near-AdS$_2$ throat, governed in part by Schwarzian modes. We study the effect of these fluctuations on the zero-frequency stress-tensor response of a near-extremal asymptotically AdS$_4$ Reissner--Nordström black brane whose transverse directions are regulated by a finite toroidal quotient. Working directly in four dimensions, we compute the leading correction in the weakly coupled Schwarzian regime $T_q\ll T\ll r_0/L^2$. At one loop, the shear perturbation couples to the $n=2$ would-be zero mode and produces a correction proportional to $T_q/T$. Together with the logarithmic correction to the entropy, this yields a finite-volume correction to $η/s$. Our calculation provides a direct four-dimensional benchmark, complementary to exact two-dimensional treatments, while keeping the asymptotic AdS$_4$ source explicit.

hep-th↗
arXiv2026

Families of Hitchin Systems in Type-D

The Coulomb branch geometry of a 4d $\mathcal{N}=2$ SCFT is encoded in the data of a complex integrable system. In class-S, this is the Hitchin System (of ADE type) on the punctured curves $C$ on which we compactified from 6d to 4d. As we vary the complex structure of $C$, these fit together to form a (nontrivial!) bundle of Hitchin systems over the moduli space of complex structures of $C$ (the ``conformal manifold'' of the family of SCFTs). We carry out that construction for type-D. Compared to the type-A case, the construction is much more complicated because of local constraints at the punctures. Those local constraints were studied in [1]. Here, we work out their implications for the global bundle of spectral (Seiberg-Witten) curves.

hep-th↗