By C.B. Wang

ISBN-10: 3642385648

ISBN-13: 9783642385643

ISBN-10: 3642385656

ISBN-13: 9783642385650

The eigenvalue densities in quite a few matrix types in quantum chromodynamics (QCD) are eventually unified during this booklet through a unified version derived from the integrable platforms. Many new density versions and loose power features are hence solved and awarded. The section transition versions together with serious phenomena with fractional power-law for the discontinuities of the loose energies within the matrix types are systematically labeled by way of a transparent and rigorous mathematical demonstration. The equipment the following will stimulate new learn instructions akin to the $64000 Seiberg-Witten differential in Seiberg-Witten thought for fixing the mass hole challenge in quantum Yang-Mills concept. The formulations and effects will gain researchers and scholars within the fields of section transitions, integrable platforms, matrix types and Seiberg-Witten theory.

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**Extra info for Application of Integrable Systems to Phase Transitions**

**Example text**

L) The matrix Aˆ n (z) is obtained from An by replacing the un−lq+s−1 and vn−lq+s (s) (s) by xn and yn respectively. One may ask whether the uN and vN functions must have such periodic behaviors. The explanation is that the string equations are applied to reorganize the wave functions, not the particles. In the momentum aspect, the parameters and the corresponding functions such as uN and vN control the wave functions of the random variables, so that the asymptotics of these functions are not directly related to the behaviors of the particles.

In this section, we discuss how to get an analytic function ω(η) with the asymptotics 12 W (η) − η1 as η → ∞ in the complex plane, where = ∂/∂η. 20) > 0. We have the following asymptotics e−V (z)/2 × − 12 V (z)+n ln z pn (z) ∼ e as z → ∞. This asymptotics gives a hint that the differential equation for the orthogonal polynomials may help us to find the ω(η). In the following, we introduce the basic Lax pair theory to construct the coefficient matrix An (z). The orthogonal polynomials satisfy a recursion formula [8], pn+1 (z) + un pn (z) + vn pn−1 (z) = zpn (z).

We finally have 1 Re π θ ω(η)dη γ3 j −1 2m = jgj j =2 q=1 j − 1 j −q−1 q+1 b a q [q/2]−μq r=0 q Θ2r+μq +1 . 10), the lemma is proved. The Θl in the above lemma can be further simplified by the recursion for some elementary integrals as described in the following. 3 For k = 0, 1, 2, . . , there are π θ e2iθ 1 − e2iθ k+ 12 0 π 0 θ eiθ 1 − e2iθ k+ 12 dθ = π , (2k + 3)i 1 dθ = −2 0 1 0 where B(·, ·) is the Euler beta function. 1 Free Energy for the One-Interval Case 49 Proof The first equation in this lemma can be easily verified by using integration by parts, π k+ 12 θ e2iθ 1 − e2iθ dθ = 0 π 1 (2k + 3)i 1 − e2iθ k+ 32 dθ = 0 π .

### Application of Integrable Systems to Phase Transitions by C.B. Wang

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