Openness of Regular Regimes of Complex Random Matrix Models

Phase diagram and topological expansion in the complex quartic random matrix model

“…Remark In fact $$\Re a_{3}, \Im a_{3}, \Re b_{3}, \Im b_{3}, \Re c_3$$ and $$\Im c_3$$ are real‐analytic functions of $$\Re \sigma$$ and $$\Im \sigma$$ for all $$\sigma \in \mathcal {O}_{3}$$. In [5], in the more general context where the external field is of even degree 2 p , $$p \in {\mathbb {N}}$$, among other things we establish the real‐analyticity of the real and imaginary parts of the end‐points for all q ‐cut regimes, $$1 \le q \le 2p-1$$, with respect to the real and imaginary parts of the complex parameters in the external field.…”

“…In fact ℜ𝑎 3 , ℑ𝑎 3 , ℜ𝑏 3 , ℑ𝑏 3 , ℜ𝑐 3 and ℑ𝑐 3 are real-analytic functions of ℜ𝜎 and ℑ𝜎 for all 𝜎 ∈  3 . In [5], in the more general context where the external field is of even degree 2𝑝, 𝑝 ∈ ℕ, among other things we establish the real-analyticity of the real and imaginary parts of the end-points for all 𝑞-cut regimes, 1 ≤ 𝑞 ≤ 2𝑝 − 1, with respect to the real and imaginary parts of the complex parameters in the external field.…”

“…The following Lemma and Lemma 3.24 are particular cases of the more general Theorem which states that for a general polynomial potential of degree p , each one of the q ‐cut critical graphs, $$1\le q \le 2p-1$$, deforms continuously with respect to the parameters in the potential (see Theorem 3 of [5]). Lemma The critical graph $$\mathcal {J}^{(1)}_{\sigma }$$ deforms continuously with respect to σ.…”

“…The following Lemma and Lemma 3.24 are particular cases of the more general Theorem which states that for a general polynomial potential of degree 𝑝, each one of the 𝑞-cut critical graphs, 1 ≤ 𝑞 ≤ 2𝑝 − 1, deforms continuously with respect to the parameters in the potential (see Theorem 3 of [5]). approach infinity along two directions 3𝜋∕4 radians apart (see (d)-(f) above).…”