Birth of a gap: Critical phenomena in 2D Coulomb gas

Technical Summary: Birth of a gap: Critical phenomena in 2D Coulomb gas

Problem Statement
This paper investigates the statistical mechanics of a two-dimensional Coulomb gas at inverse temperature $\beta = 2$, modeled as a determinantal point process of $n$ particles in the complex plane. The study focuses on a specific family of radially symmetric external potentials $Q(z) = q(|z|)$ where the equilibrium measure's density vanishes on a circle of radius $r^* > 0$ strictly inside the droplet (the support of the equilibrium measure). This configuration represents a critical regime where a spectral gap is about to emerge in the bulk of the droplet, separating it into either an annulus or a disk with a hole. The authors aim to characterize the asymptotic behavior of the system as $n \to \infty$ in this critical setting, specifically analyzing the partition function and local correlation functions near the radius $r^*$.

Methodology
The analysis relies on the determinantal structure of the point process at $\beta=2$, where the correlation kernel $K_n$ is constructed from weighted monic orthogonal polynomials $p_{j,n}(z)$. The methodology proceeds through several rigorous analytical steps:

1. Asymptotic Expansion of Norms: The authors analyze the norms $\|p_{j,n}\|$ of the orthogonal polynomials. They employ the Laplace method on the integral representation of these norms. A crucial aspect is the identification of a critical index $j^* \approx n\tau^*$ corresponding to the radius $r^*$. Near this critical index, the standard Gaussian approximation fails because the second derivative of the effective potential vanishes. Instead, the authors perform a higher-order Taylor expansion (up to fifth order) of the potential around the critical point, leading to a non-Gaussian scaling.
2. Scaling and Coordinate Transformation: To handle the vanishing derivatives, the authors introduce a specific scaling for the radial distance from $r^*$, proportional to $n^{-1/4}$. They define a change of coordinates relating the index $j$ to this radial distance, utilizing the implicit function theorem to relate the parameter $\tau = j/n$ to the radius $r_\tau$.
3. Partition Function Expansion: The logarithm of the partition function $Z_n$ is expressed as a sum of the logarithms of the polynomial norms. The authors split this sum into three regimes: indices far from the critical index (where standard Laplace method applies), indices near the critical index (where the new $n^{1/4}$ scaling dominates), and the boundary terms. The Euler-Maclaurin formula is used to convert these sums into integrals, carefully handling the singularities introduced by the critical point.
4. Local Correlation Analysis: For the local kernel near $r^*$, the authors perform a double scaling limit. They rescale the coordinates $z_k = r^* + \xi_k / (\gamma_Q n)^{1/4}$ and analyze the limit of the kernel $K_n$. This involves evaluating the asymptotic behavior of the orthogonal polynomials in the critical regime and summing the resulting series using the Euler-Maclaurin formula again.

Key Contributions and Results

• Novel Term in Partition Function Expansion: The primary result regarding the partition function is the derivation of a large-$n$ asymptotic expansion for $\log(Z_n)$. Unlike previous results for non-critical potentials (which typically follow an expansion of the form $C_0 n^2 + C_1 n \log n + C_2 n + C_3 \log n + C_4$), this critical case features a novel term of order $n^{1/4}$.

  • The expansion is given by:
    $$\log \left( \frac{Z_n}{n! (2\pi)^n} \right) = -n^2 I_Q[\mu_Q] - \frac{1}{2}n \log n + n \left[ \frac{\log(\pi/2)}{2} - \frac{1}{2}E_Q[\mu_Q] \right] + n^{1/4} \frac{\kappa_Q}{2} \int_{-\infty}^{\infty} x^2 \log \left( \frac{x^2}{\sqrt{4\pi} P(x^4)} \right) dx + F_Q + o(1)$$
  • Here, $P(y)$ is a specific "Pearcey-like" integral function, and $\kappa_Q$ depends on the second derivative of the Laplacian of the potential at $r^*$. The constant term $F_Q$ is also derived explicitly, generalizing previous results to account for the singularity at $r^*$.

• New Universal Limiting Kernel: The authors derive the local correlation kernel in the vicinity of $r^*$ under the double scaling limit where the radial distance is of order $n^{-1/4}$.

  • This limit yields a new universal kernel $K^*(\xi_1, \xi_2)$, which differs from the standard Ginibre kernel found in the bulk of non-critical Coulomb gases.
  • The kernel is expressed as an integral involving the Pearcey-like function $P$ and its derivatives. It governs the local point process at the critical radius.
  • The authors provide the explicit form of the first correction term of order $n^{-1/4}$, noting that while the leading term is universal, the correction depends on the specific potential $Q$.

• Mean Level Spacing: A direct consequence of the scaling analysis is the determination of the mean level spacing of particles near $r^*$. The authors prove that the average distance between particles in the vicinity of the critical radius scales as $n^{-1/4}$, which is distinct from the $n^{-1/2}$ scaling typical of the bulk.

• Generalization to Higher Criticality: The paper formulates conjectures for potentials where the vanishing of the Laplacian is of higher order (i.e., the first non-vanishing derivative is of order $2m$). In such cases, the authors conjecture that the partition function expansion will feature a term of order $n^{1/(2m+2)}$ and the local kernel will scale with $n^{-1/(2m+2)}$, involving generalized Pearcey-like integrals.

Significance
The paper claims significance in identifying and characterizing a new universality class for 2D Coulomb gases associated with the "birth of a gap" in the bulk.

• Theoretical Novelty: The appearance of the $n^{1/4}$ term in the free energy expansion is highlighted as a new phenomenon in the literature, distinct from the surface tension terms ($n^{1/2}$) or standard bulk terms. The authors suggest this term is related to the criticality of the potential and the specific geometry of the emerging gap.
• New Kernel: The derivation of the kernel $K^*$ provides a new mathematical object describing local correlations in critical 2D systems, distinct from the well-known Ginibre ensemble.
• Critical Scaling: The work establishes that the degree of criticality of the potential (how many derivatives vanish at the gap formation point) directly dictates the scaling of the mean level spacing and the order of the sub-leading terms in the partition function.

The authors maintain a modest tone regarding the scope, noting that their results are currently restricted to radially symmetric potentials and $\beta=2$. They acknowledge that extending these results to broken radial symmetry or other values of $\beta$ remains an open question, and they propose the study of the interpolation between the Ginibre bulk and this critical kernel as a direction for future work.