Multidimensional Heine Distribution

• Multidimensional Heine distribution is a product-form law on ℤ₊^k derived as the discrete limit of the q-multinomial distribution, featuring q-Pochhammer factors.
• It appears in dual contexts: as independent Heine marginals in discrete models and as fluctuation laws near spectral outposts in two-dimensional Coulomb gas settings.
• Key insights include the use of orthogonal polynomial asymptotics, local limit theorems, and capacity-based parameters that universally determine fluctuation regimes.

Searching arXiv for the cited papers and closely related work on Heine distributions and Coulomb-gas fluctuations. Multidimensional Heine distribution is used in recent arXiv literature in two related but non-identical ways. In (Vamvakari, 2024), “multiple Heine distribution,” “multivariate Heine distribution,” and “multidimensional Heine distribution” are synonymous and denote a product-form law on $\mathbb{Z}_+^k$ obtained as the discrete limit of the $q$-multinomial distribution of the first kind. In (Ameur et al., 2024), the Heine law appears in the normal matrix model at $\beta=2$ as the asymptotic law of the particle count near a spectral outpost in a fully two-dimensional Coulomb gas; in the spectral-gap setting, the same work identifies a “multi-Heine” structure through differences of independent Heine variables. In both settings, $q$-Pochhammer factors are structurally central, but the underlying mechanisms are different: discrete $q$-multinomial limits in one case, and orthogonal-polynomial bifurcation plus partition-function asymptotics in the other.

1. Terminology and defining formulas

The univariate Heine distribution used in (Ameur et al., 2024) is the law $\operatorname{He}(\theta,q)$, with $\theta>0$ and $0$X\in\mathbb{Z}_+=\{0,1,2,\dots\}$ such that

$$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$

where

$q$0

Its cumulant generating function is

$q$1

The expectation is

$q$2

In (Vamvakari, 2024), the univariate Heine distribution of the first kind is parameterized by $q$3, with $q$4 and $q$5, and has probability mass function

$q$6

where

$q$7

and

$q$8

The normalizing constant is

$q$9

The multidimensional, or multiple, Heine distribution in (Vamvakari, 2024) is defined on $\beta=2$0, $\beta=2$1, by the product-form joint probability mass function

$\beta=2$2

Thus $\beta=2$3 are independent. This product structure is the defining feature of the multiple Heine law in that paper (Vamvakari, 2024).

2. Discrete origin in the $\beta=2$4-multinomial distribution of the first kind

The multiple Heine distribution arises in (Vamvakari, 2024) as the discrete limit of the $\beta=2$5-multinomial distribution of the first kind. The model is given through a “chain composite failures” construction: if $\beta=2$6 is the number of successes of kind $\beta=2$7 in $\beta=2$8 independent Bernoulli trials, then the success probability of kind $\beta=2$9 at trial $q$0 is

$q$1

with $q$2 (or $q$3 at the level of model definition).

The $q$4-multinomial coefficient used there is

$q$5

and the joint probability mass function is

$q$6

for $q$7 and $q$8.

The limiting regime is $q$9 with $q$0 fixed in $q$1 and fixed $q$2, together with the parameter correspondence

$q$3

Under this limit,

$q$4

Equivalently,

$q$5

The same paper establishes local limit theorems. For the $q$6-multinomial of the first kind, the assumptions are $q$7 fixed and

$q$8

with $q$9. For the multiple Heine distribution, the local limit regime is $\operatorname{He}(\theta,q)$0. The continuous limit is a multivariate Stieltjes–Wigert type distribution built from the univariate density

$\operatorname{He}(\theta,q)$1

The centering and scaling are performed after deformation by $\operatorname{He}(\theta,q)$2. For the multiple Heine components,

$\operatorname{He}(\theta,q)$3

The proofs use pointwise convergence in a “$\operatorname{He}(\theta,q)$4-analogous sense” and a $\operatorname{He}(\theta,q)$5-Stirling type asymptotic for $\operatorname{He}(\theta,q)$6 with an explicit $\operatorname{He}(\theta,q)$7 term; the final local limit approximations are stated in the “$\operatorname{He}(\theta,q)$8” sense, without explicit uniform error bounds for the probability mass functions (Vamvakari, 2024).

3. Two-dimensional Coulomb-gas formulation

In (Ameur et al., 2024), the Heine distribution is embedded in the Coulomb gas, or normal matrix model, at $\operatorname{He}(\theta,q)$9. Let $\theta>0$0 be a confining potential that is lower semicontinuous, finite on a set of positive capacity, and satisfies

$\theta>0$1

For $\theta>0$2 particles $\theta>0$3, the Gibbs measure is

$\theta>0$4

where $\theta>0$5 and $\theta>0$6 is the partition function.

The equilibrium measure $\theta>0$7 is the unique minimizer of

$\theta>0$8

and its support $\theta>0$9 is the droplet. If $0

$0

with $0

$0

and the coincidence set is

$0

One always has $0

The outpost geometry is defined under the assumptions that $0$X\in\mathbb{Z}_+=\{0,1,2,\dots\}$0, and $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$1 is the normalized exterior map

$X\in\mathbb{Z}_+=\{0,1,2,\dots\}$2

Fix $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$3 and define

$X\in\mathbb{Z}_+=\{0,1,2,\dots\}$4

The bounded domain with boundary $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$5 is the ring-shaped gap $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$6.

The potential $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$7 is an outpost potential when: (i) $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$8 is real-analytic near $X\in\mathbb{Z}_+=\{0,1,2,\dots\}$9; (ii) the coincidence set is $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$0; (iii) there exists $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$1 holomorphic in $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$2 and a real constant $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$3 such that

$$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$4

has boundary values $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$5 on $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$6 and on $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$7, where $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$8 is harmonic on $$P(X=j)=\big[(-\theta;q)_\infty\big]^{-1}\frac{q^{j(j-1)/2}\theta^j}{(q;q)_j},\qquad j=0,1,2,\ldots,$$9 with $q$00 and $q$01. The function $q$02 extends holomorphically to $q$03 and is normalized by $q$04; one also defines $q$05 on $q$06, with $q$07 on $q$08.

Under these assumptions, the outpost $q$09 lies in $q$10, has zero area but positive logarithmic capacity, and affects mesoscopic fluctuations. The particle count near the outpost is defined by choosing $q$11 with $q$12 in a neighborhood of $q$13 and $q$14 in a neighborhood of $q$15, and setting

$q$16

The analysis uses a mesoscopic boundary layer of width

$q$17

together with ring neighborhoods

$q$18

for $q$19 large and $q$20 small fixed (Ameur et al., 2024).

4. Heine law near a spectral outpost

The central fluctuation result of (Ameur et al., 2024) is a Heine law for the outpost count. Let $q$21, $q$22, and let $q$23 be the constant in the decomposition

$q$24

on the gap $q$25 with boundary values $q$26. Then, as $q$27,

$q$28

with

$q$29

More precisely, if $q$30 and

$q$31

then uniformly for $q$32,

$q$33

The parameter $q$34 is universal and equals the square of the capacity ratio. The parameter $q$35 incorporates the harmonic datum $q$36 determined by the boundary values of $q$37 on $q$38 and $q$39 via the Dirichlet decomposition. In the quasi-harmonic case with $q$40 along $q$41, $q$42, one finds

$q$43

so that

$q$44

which recovers the radially symmetric result.

The Heine law is identified through orthogonal polynomial norms in the bifurcation window

$q$45

For monic orthogonal polynomials $q$46 in the perturbed weight $q$47, one has

$q$48

and, more precisely,

$q$49

uniformly for $q$50. The auxiliary identities

$q$51

convert the two-peak asymptotics into the $q$52-Pochhammer product appearing in the Heine cumulant generating function.

An equivalent formulation uses partition function ratios. For the perturbation

$q$53

if $q$54 denotes the perturbed partition function, then

$q$55

In the bifurcation window, the two-term structure of $q$56 produces a factor of the form $q$57 inside the logarithm, and summation over $q$58 yields a $q$59-Pochhammer factor $q$60 (Ameur et al., 2024).

5. Universality, spectral gaps, and fluctuation decomposition

The outpost theorem in (Ameur et al., 2024) is universal in a precise sense. Under the outpost assumptions—analytic boundary, $q$61 a level set of $q$62, and the compatibility condition—the asymptotic distribution of $q$63 depends only on:

1. the capacity ratio $q$64, and
2. the constant $q$65 in the decomposition $q$66 on $q$67 with boundary values $q$68.

In particular, it is independent of the global geometry of $q$69 and of the detailed shape of $q$70 and $q$71 beyond these two numerical invariants. The setting is genuinely two-dimensional: non-radial potentials, nontrivial conformal maps $q$72, and planar ring domains $q$73 replace the radially symmetric multi-annulus geometry.

A second fluctuation regime arises for disconnected droplets separated by a ring-shaped spectral gap. If $q$74 has two components separated by a gap $q$75 with boundaries $q$76 and $q$77, let $q$78 be the equilibrium mass of the inner component and $q$79 the fractional part. Then there exist independent Heine variables $q$80 such that

$q$81

with

$q$82

$q$83

and

$q$84

uniformly for $q$85. Consequently, the difference

$q$86

has a discrete normal distribution:

$q$87

This law depends on $q$88 through $q$89, so it is asymptotically oscillatory in $q$90.

For general smooth linear statistics in the ring case, one writes

$q$91

Then the cumulant generating function splits as

$q$92

where $q$93 is Gaussian with mean

$q$94

and variance

$q$95

Here $q$96 is a fixed smooth extension of $q$97 near $q$98, $q$99 is the Neumann jump operator across $\beta=2$00, and $\beta=2$01 is the Poisson modification of $\beta=2$02. Thus the fluctuation field decomposes into an $\beta=2$03-independent Gaussian part and an independent oscillatory discrete Gaussian part (Ameur et al., 2024).

6. Comparisons, interpretation, and limits of current results

The two cited papers use closely related $\beta=2$04-structures but different notions of “multidimensional.” In (Vamvakari, 2024), multidimensionality means a $\beta=2$05-variate law on $\beta=2$06 with independent Heine marginals. In (Ameur et al., 2024), multidimensionality refers to the fully two-dimensional Coulomb-gas geometry: a non-radial potential on $\beta=2$07, a genuine planar outpost $\beta=2$08 in $\beta=2$09, and fluctuation parameters determined by capacities and harmonic Dirichlet data rather than by one-dimensional radii alone.

The Coulomb-gas results are presented as a generalization of earlier radially symmetric multi-annulus results. The present setting replaces concentric circles by Jordan curves, and the parameters are described through

$\beta=2$10

The paper emphasizes that the outpost law depends only on the conformal modulus, through the capacity ratio, and on the harmonic datum $\beta=2$11, not on global geometric details (Ameur et al., 2024).

The heuristic contrast with one-dimensional “birth of a cut” is also explicit there. In the planar ring-gap or outpost geometry, orthogonal polynomials develop two competing peaks near $\beta=2$12 and $\beta=2$13, and the ratio of weights between consecutive degrees is geometric with common ratio

$\beta=2$14

Summing the resulting transfer contributions produces $\beta=2$15-Pochhammer factors characteristic of Heine distributions. By contrast, the paper states that in one-dimensional “birth of a cut,” nearby eigenvalues strongly interact and no such $\beta=2$16-product arises.

Several boundary cases remain conjectural or only partially developed. Outposts with zero capacity, such as points, are conjectured to attract no particles asymptotically, so the count should tend to $\beta=2$17 in distribution. For several disjoint gaps or outposts, more general multivariate joint laws are expected to factor asymptotically as products of Heine factors under suitable separation, as suggested by the free-energy surmise, but a general theorem for multiple outposts is not proved in (Ameur et al., 2024). This suggests that the multiple Heine distribution of (Vamvakari, 2024) provides an exact product-model counterpart to a factorization pattern that, in the Coulomb-gas setting, currently remains at the level of expected asymptotic structure rather than a complete theorem.