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Large-N Transitions and Critical Phenomena

In: Application of Integrable Systems to Phase Transitions

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  • C. B. Wang

    (Institute of Analysis)

Abstract

The bifurcation transition models discussed in the last chapter can be extended to large-N transitions, which will be explained in this chapter based on hypergeometric-type differential equations and the double scaling method. The singular values of the hypergeometric-type differential equation are related to the elliptic functions that are the fundamental mathematical tools for studying the vertex models in statistical physics. The double scaling method can connect the string system to the soliton system. Different transitions, or discontinuities, will be discussed in this chapter, especially the odd-order transitions, such as first-, third- and fifth-order transitions, which can be formulated by using the density models. The second-order divergences (critical phenomena) that are usually discussed in physics by using renormalization methods can be obtained by considering the derivatives of the logarithm of the partition function in the original potential parameter direction and using the Toda lattice. The third-order divergence for the planar diagram model is investigated in association with the critical phenomenon and double scaling. The fourth-order discontinuity is studied by using the analytic properties of the integrable system.

Suggested Citation

  • C. B. Wang, 2013. "Large-N Transitions and Critical Phenomena," Springer Books, in: Application of Integrable Systems to Phase Transitions, edition 127, chapter 0, pages 75-106, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-38565-0_4
    DOI: 10.1007/978-3-642-38565-0_4
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