Phase Diagram and Topological Expansion in the Complex Quartic Random Matrix Model

Openness of Regular Regimes of Complex Random Matrix Models

“…. , 𝑛, is given by the probability distribution Formulae (5), (6) are well known for real polynomial potentials 𝑉 (𝑧) of even degree (see, e.g., [6]), and their proof for a complex 𝑉 (𝑧) goes through without any change. By Heine's formula (see e.g.…”

“…By Heine's formula (see e.g. [23]) the multiple integral in (5) is, up to a multiplicative constant, the determinant of the Hankel matrix 𝐻 𝑛 [𝑤] := {𝑤 𝑗+𝑘 } 𝑘, 𝑗=0,...,𝑛 , where 𝑤(𝑥; 𝑡 𝑡 𝑡) ≡ exp[−𝑛𝑉 (𝑥; 𝑡 𝑡 𝑡)] and 𝑤 ℓ is the ℓ-th moment of the weight 𝑤(𝑥; 𝑡 𝑡 𝑡). Correspondingly, one can also consider the system of monic orthogonal polynomials {𝑃 𝑛 (𝑧; 𝑡 𝑡 𝑡)} 𝑛∈Z ≥0 satisfying ∫ Γ 𝑃 𝑛 (𝑧; 𝑡 𝑡 𝑡)𝑧 𝑘 𝑤(𝑧; 𝑡 𝑡 𝑡)𝑑𝑧 = 0 , for 𝑘 = 0, 1, .…”

“…, 𝑛 − 1. (7) The connection of this system of orthogonal polynomials and the partition function (5) can be seen as follows: the orthogonal polynomial of degree 𝑛 exists and is unique if the partition function Z 𝑛 (𝑡 𝑡 𝑡), or the 𝑛 × 𝑛 Hankel determinant det 𝐻 𝑛 [𝑤], is nonzero. Indeed, the existence follows from the explicit formula (8) 𝑃 𝑛 (𝑧; 𝑡 𝑡 𝑡) ≡ 𝑃 𝑛 (𝑧) = 1 det 𝐻 𝑛 [𝑤] det…”

“…In [2] the phase diagrams for a variety of choices of integration contours for this model have been presented. In [5] the particular case of admissible contours that approach ∞ along the real axis was considered and the phase diagram (as shown in Figure 1 ) was proven. Using the explicit formulae for the end-points of 𝐽 𝜎 and zeros of ℎ(𝑧; 𝜎) in the one-cut case, one can easily find that the non-generic parameter values corresponding to case (a) above are only 𝜎 = ±i √ 12, for which the points ∓𝑧 0 (zeros of ℎ(𝑧; 𝜎)) coincide with the endpoints ±𝑏 1 [5,2].…”

“…In [5] the particular case of admissible contours that approach ∞ along the real axis was considered and the phase diagram (as shown in Figure 1 ) was proven. Using the explicit formulae for the end-points of 𝐽 𝜎 and zeros of ℎ(𝑧; 𝜎) in the one-cut case, one can easily find that the non-generic parameter values corresponding to case (a) above are only 𝜎 = ±i √ 12, for which the points ∓𝑧 0 (zeros of ℎ(𝑧; 𝜎)) coincide with the endpoints ±𝑏 1 [5,2]. The non-generic points on the boundaries labeled by 𝛾 1 and 𝛾 2 represent the 𝜎 values for which the zeros of ℎ(𝑧; 𝜎) hit the support of the equilibrium measure (see Figure 2).…”