By C.B. Wang

The eigenvalue densities in a variety of matrix versions in quantum chromodynamics (QCD) are eventually unified during this booklet through a unified version derived from the integrable structures. Many new density versions and loose power features are hence solved and offered. The section transition versions together with serious phenomena with fractional power-law for the discontinuities of the unfastened energies within the matrix types are systematically categorised via a transparent and rigorous mathematical demonstration. The equipment right here will stimulate new study instructions reminiscent of the $64000 Seiberg-Witten differential in Seiberg-Witten conception for fixing the mass hole challenge in quantum Yang-Mills concept. The formulations and effects will gain researchers and scholars within the fields of part transitions, integrable structures, matrix types and Seiberg-Witten theory.

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**Example text**

Since det Dn = hn hn−1 and vn = hn / hn−1 , there is det Fn = 2mt2m an,n−1 z2m−2 (1 + O(z−1 )). 40), we have det Fn (z) = 2mnt2m z2m−2 1 + O z−1 . 70) is proved. 4 Density Models 1 j 1 1 For the density on one interval, denote z/n 2m , tj /n1− 2m , xn /n 2m , and yn /n m by η, gj , a, and b2 respectively, where b > 0. The a and b will be called cen1 ter and radius parameters respectively in the later discussions. Let αn = n 2m α, 1 and then yn αn−1 = n 2m (b2 α −1 ), where α = (η − a + (η − a)2 − 4b2 )/2, and 38 2 Densities in Hermitian Matrix Models b2 α −1 = (η − a − (η − a)2 − 4b2 )/2.

The string equations can be applied to find the critical point for the transition from the parameter bifurcation, and the bifurcation clearly separates the different phases for analyzing the free energy. Based on the expansion method for elliptic integrals, the third-order bifurcation transition for the Hermitian matrix model with a general quartic potential is discussed in this chapter by applying the nonlinear relations obtained from the string equations. The density on multiple disjoint intervals for higher degree potential and the corresponding free energy are discussed in association with the SeibergWitten differential.

45) has either a factor xn or tj with odd j . 44). For the density on multiple disjoint intervals, consider 0 (1) −yn Jˆn(l) = 1 0 (1) · · · (l) −yn z − xn 1 (l) . 48) tr Jˆn(l) I = Jˆn(l) + det Jˆn(l) Jˆn(l) . We can transform tj (j = 1, . . , 2m) into a new set of parameters tj (j = 1, . . 49) q=0 where each ms (s = 0, . . , l − 1) is the largest integer such that s + lms ≤ 2m − 1. In fact, by expanding the above expression in terms of z and comparing the coefficients j −1 , we can get a upper triangle matrix T with V (z) = 2m 2m so that T2m t = j =1 j tj z T T t where t = (t1 , 2t2 , .