Muttalib–Borodin Determinants

• Muttalib–Borodin determinants are generalized Hankel determinants that model partition functions in complex random matrix ensembles using biorthogonal polynomials.
• They incorporate an extra interaction parameter θ to capture two-body interactions and handle singular weight structures like Fisher–Hartwig singularities.
• This framework enables precise asymptotic analysis, including equilibrium measure derivations, central limit theorems, and rigidity estimates in advanced random matrix models.

The Muttalib–Borodin determinant is a core object in the generalization of Hankel determinants arising in random matrix theory and biorthogonal polynomial ensembles. It provides a unified framework connecting determinantal processes with two-body interactions beyond traditional Vandermonde factors and underpins the theory of Muttalib–Borodin (MB) ensembles, a class of biorthogonal ensembles characterized by an extra interaction parameter $\theta>0$. MB determinants play a central role in asymptotic analysis of partition functions, characteristic polynomials, correlation functions, and global rigidity features in random particle systems, particularly in the presence of singular weights such as those with Fisher–Hartwig structure (Charlier, 2021). The theory combines techniques from Toeplitz/Hankel determinant asymptotics, potential theory, and nonlinear steepest-descent on Riemann–Hilbert (RH) problems.

1. Definition and Equivalent Representations

Let $\theta>0$, $0<a<b<+\infty$, and $w:[a,b]\rightarrow\mathbb{C}$ a sufficiently regular weight function. The $n\times n$ Muttalib–Borodin determinant is

$$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$

An alternative matrix determinant form is

$$D_n(w) = \det\left[\int_{a}^{b} x^{k + j\theta} w(x)\,dx\right]_{j,k=0}^{n-1}.$$

These generalize Hankel determinants, reducing to them for $\theta=1$. In the context of biorthogonal ensembles, there exist two sequences of monic polynomials $\{p_j\}_{j\geq0}$ and $\{q_j\}_{j\geq0}$, of degree $\theta>0$0, characterized by biorthogonality

$\theta>0$1

for $\theta>0$2, $\theta>0$3. The normalization constants satisfy $\theta>0$4, yielding

$\theta>0$5

for any $\theta>0$6 (Charlier, 2021).

2. Fisher–Hartwig Singularities and Weight Structure

The Fisher–Hartwig weight on $\theta>0$7 is written as $\theta>0$8, where $\theta>0$9 is real-analytic near $0<a<b<+\infty$0 and

$0<a<b<+\infty$1

with $0<a<b<+\infty$2 and $0<a<b<+\infty$3. The points $0<a<b<+\infty$4 and $0<a<b<+\infty$5 correspond to "hard-edge" root-type singularities, $0<a<b<+\infty$6 are interior root/jump singularities. This generalizes the classical Fisher–Hartwig setting for Toeplitz and Hankel determinants and arises naturally in non-intersecting particle models and deformed random matrix potentials (Charlier, 2021).

3. Logarithmic Energy Equilibrium Problem

In the large-$0<a<b<+\infty$7 limit, the leading-order asymptotics are controlled by a discrete-to-continuous equilibrium problem: $0<a<b<+\infty$8 The minimizer $0<a<b<+\infty$9 is absolutely continuous on $w:[a,b]\rightarrow\mathbb{C}$0, $w:[a,b]\rightarrow\mathbb{C}$1, and the density satisfies

$w:[a,b]\rightarrow\mathbb{C}$2

for some constant $w:[a,b]\rightarrow\mathbb{C}$3. The support $w:[a,b]\rightarrow\mathbb{C}$4 is described via a conformal mapping $w:[a,b]\rightarrow\mathbb{C}$5, where $w:[a,b]\rightarrow\mathbb{C}$6 are fixed by endpoint conditions (Charlier, 2021). The equilibrium density admits the representation

$w:[a,b]\rightarrow\mathbb{C}$7

where $w:[a,b]\rightarrow\mathbb{C}$8 is the functional inverse of $w:[a,b]\rightarrow\mathbb{C}$9 off $n\times n$0.

4. Large $n\times n$1 Asymptotics: Full Expansion

For $n\times n$2, Charlier (Charlier, 2021) established the asymptotic expansion

$n\times n$3

where $n\times n$4. The explicit constants are: $n\times n$5 $n\times n$6 is $n\times n$7-independent, in closed form only for $n\times n$8. These constants depend continuously on singularity placement and exponents, with error uniform for sufficiently separated singularities (Charlier, 2021).

5. Corollaries: Central Limit Theorems, Rigidity, and Characteristic Polynomials

Several statistical consequences are immediate for the MB ensemble:

• The expected number of points left of $n\times n$9 obeys $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$0 and variance $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$1.
• The real and imaginary parts of the (logarithm of the) characteristic polynomial concentrate, with $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$2.
• Joint multivariate central limit theorems hold for collections of position and logarithmic polynomial observables, upon suitable normalization.
• Bulk rigidity: With high probability, the maximal deviation of empirical counting function from its expectation is $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$3, and the particle positions concentrate around their classical locations with fluctuations $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$4 (Charlier, 2021).

6. Asymptotic Methodology: Riemann–Hilbert Problem and Steepest Descent

The primary technical tool is a $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$5 vector-valued RH problem encoding the monic polynomial $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$6 and its $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$7-Cauchy transform. The Deift–Zhou nonlinear steepest descent is adapted to this setting:

• Introduce $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$8 and $$D_n(w) = \frac{1}{n!}\int_{a}^{b}\cdots\int_{a}^{b} \prod_{1\leq j<k\leq n}(x_k-x_j)(x_k^{\theta}-x_j^{\theta}) \prod_{j=1}^n w(x_j)\,dx_j.$$9 functions from the equilibrium measure.
• Open lenses deforming the jump contours.
• Construct local parametrices at endpoints $$D_n(w) = \det\left[\int_{a}^{b} x^{k + j\theta} w(x)\,dx\right]_{j,k=0}^{n-1}.$$0 (Bessel form) and at interior Fisher–Hartwig points (confluent hypergeometric form).
• Employ a global parametrix using the conformal map $$D_n(w) = \det\left[\int_{a}^{b} x^{k + j\theta} w(x)\,dx\right]_{j,k=0}^{n-1}.$$1.
• Employ a "shifted" small-norm RH analysis to extract precise expansions for normalization constants $$D_n(w) = \det\left[\int_{a}^{b} x^{k + j\theta} w(x)\,dx\right]_{j,k=0}^{n-1}.$$2, which via the recursion yield $$D_n(w) = \det\left[\int_{a}^{b} x^{k + j\theta} w(x)\,dx\right]_{j,k=0}^{n-1}.$$3 (Charlier, 2021).

This approach builds on methods for Toeplitz/Hankel determinants with Fisher–Hartwig singularities, as developed in [Deift–Its–Krasovsky 2011], and is extended to the two-body interaction setting of Muttalib–Borodin ensembles.

7. Related Models and Applications

Muttalib–Borodin determinants naturally generalize the partition functions of classical and deformed random matrix ensembles, including those with spectral singularities and inhomogeneous/critical statistics. They play a central role in the study of global and mesoscopic statistics for random matrices and serve as the foundational partition functions for MB ensembles. The large-$$D_n(w) = \det\left[\int_{a}^{b} x^{k + j\theta} w(x)\,dx\right]_{j,k=0}^{n-1}.$$4 expansions and rigidity results have direct application in random matrix theory, statistical mechanics, and asymptotic combinatorics involving intricate singularities in the underlying weights. The analysis of MB determinants is also instrumental in establishing universality at the soft edge, in phase transition studies, and strongly influences the development of integrable probability techniques for ensembles with non-trivial interaction and singular structure (Charlier, 2021).