q-Racah Probability Measure

• The q-Racah probability measure is defined as the normalized discrete orthogonality weight for q-Racah polynomials on a q-quadratic lattice, ensuring finite support and proper normalization.
• It functions as the spectral measure for finite Jacobi matrices, underpinning one-body weights in discrete quantum mechanics, XX spin chains, and other integrable systems.
• The measure extends to diverse frameworks including orthogonal polynomial ensembles, exceptional and multi-indexed deformations, which enhances its role in both algebraic and probabilistic applications.

The q-Racah probability measure is, in its most standard sense, the normalized finite discrete orthogonality measure for the q-Racah polynomials on a q-quadratic lattice. In adjacent literatures, the same underlying weight appears as the one-body weight of the q-Racah orthogonal polynomial ensemble, as the squared ground-state vector of a finite Jacobi operator in discrete quantum mechanics, and as the endpoint spectral measure of certain exactly solvable XX spin chains. The central structural feature is finiteness: after the truncation $\gamma=q^{-N-1}$, the measure is supported on $N+1$ points, typically written either as lattice indices $x=0,1,\dots,N$ or as spectral points $q^{-x}+\gamma\delta q^{x+1}$ (Dzhamay et al., 2019, Crampe et al., 13 Dec 2025, Odake et al., 2011).

1. Finite q-Racah orthogonality measure

In the standard one-variable formulation, the q-Racah weight is a finite discrete weight on $x=0,1,\dots,M$, with $\gamma=q^{-M-1}$, quadratic-lattice variable

$\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$

and one-body weight

$$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$

If $P_n$ denotes the monic orthogonal polynomial of degree $n$ in the variable $N+1$0, then

$N+1$1

This is the finite q-Racah orthogonality relation in the form used for the q-Racah ensemble (Dzhamay et al., 2019).

A second standard parametrization is the discrete-quantum-mechanical one. There the support is again finite, $N+1$2, and the orthogonality weight is the squared ground-state vector

$N+1$3

The associated q-Racah polynomials satisfy

$N+1$4

Since $N+1$5, normalization gives a genuine probability measure whenever the weight is nonnegative: $N+1$6 The same paper identifies this parametrization with conventional q-Racah notation through

$N+1$7

with

$N+1$8

This shows that the orthogonality measure is standard q-Racah in substance, even when written as $N+1$9 (Odake et al., 2011).

Object	Support	Weight data
Standard q-Racah orthogonality measure	$x=0,1,\dots,N$0 or $x=0,1,\dots,N$1	$x=0,1,\dots,N$2
rdQM form	$x=0,1,\dots,N$3	$x=0,1,\dots,N$4
q-Racah ensemble	$x=0,1,\dots,N$5	Vandermonde$x=0,1,\dots,N$6$x=0,1,\dots,N$7

2. Recurrence, Jacobi matrices, and spectral meaning

The finite q-Racah measure is equivalently the spectral measure of a finite Jacobi matrix. In the recurrence normalization used for q-Racah-type XX chains,

$x=0,1,\dots,N$8

with

$x=0,1,\dots,N$9

$q^{-x}+\gamma\delta q^{x+1}$0

Under the finite truncation

$q^{-x}+\gamma\delta q^{x+1}$1

the support becomes

$q^{-x}+\gamma\delta q^{x+1}$2

This is the finite q-Racah lattice underlying the measure (Crampe et al., 13 Dec 2025).

In discrete quantum mechanics, the same structure is encoded by a tridiagonal Hamiltonian

$q^{-x}+\gamma\delta q^{x+1}$3

with eigenvectors $q^{-x}+\gamma\delta q^{x+1}$4. The orthogonality measure is therefore exactly the square of the ground state, and the q-Racah polynomials are the polynomial part of the eigenbasis (Odake et al., 2011).

This spectral interpretation becomes especially concrete in the inhomogeneous XX chain of q-Racah type. There the one-excitation Hamiltonian is a Jacobi matrix with entries

$q^{-x}+\gamma\delta q^{x+1}$5

and spectrum

$q^{-x}+\gamma\delta q^{x+1}$6

The endpoint spectral measure is

$q^{-x}+\gamma\delta q^{x+1}$7

and these $q^{-x}+\gamma\delta q^{x+1}$8 are exactly the normalized q-Racah orthogonality weights. In this sense, the q-Racah probability measure is the physically relevant probability distribution of endpoint spectral mass in the one-excitation sector (Crampe et al., 13 Dec 2025).

3. Positivity and normalization regimes

The finite q-Racah measure is algebraically defined by orthogonality, but positivity is parameter-dependent. For the q-Racah ensemble, a concrete admissibility regime is

$q^{-x}+\gamma\delta q^{x+1}$9

$x=0,1,\dots,M$0

Under these hypotheses the q-Racah ensemble weights are nonnegative on the whole configuration space (Dzhamay et al., 2019).

In the rdQM parametrization, a positive range used for the ordinary q-Racah system is

$x=0,1,\dots,M$1

The corresponding finite weight $x=0,1,\dots,M$2 is then positive on $x=0,1,\dots,M$3 (Odake et al., 2011).

For q-Racah spin chains, positivity is reformulated in Jacobi-matrix language. The physically relevant condition is

$x=0,1,\dots,M$4

so that $x=0,1,\dots,M$5 is real. In the first persymmetric q-Racah regime,

$x=0,1,\dots,M$6

the paper states that positivity of $x=0,1,\dots,M$7 requires

$x=0,1,\dots,M$8

This does not change the nature of the measure; it restricts the admissible parameter set (Crampe et al., 13 Dec 2025).

A recurrent point in the literature is that many papers present the orthogonality weight rather than a normalized probability law. The probability measure is then obtained by dividing by total mass. In the ordinary q-Racah case this normalization is immediate because $x=0,1,\dots,M$9, and in the rdQM notation it is encoded by the factor $\gamma=q^{-M-1}$0 (Odake et al., 2011).

4. Orthogonal polynomial ensembles and integrable probability

The q-Racah probability measure also appears as the one-body input of a discrete orthogonal polynomial ensemble. On ordered $\gamma=q^{-M-1}$1-tuples

$\gamma=q^{-M-1}$2

the q-Racah ensemble is

$\gamma=q^{-M-1}$3

with

$\gamma=q^{-M-1}$4

Thus the pair interaction is the squared Vandermonde in the quadratic-lattice variable, not directly in $\gamma=q^{-M-1}$5. The orthogonal polynomials associated with this weight are the q-Racah orthogonal polynomials (Dzhamay et al., 2019).

A distinguished observable is the one-interval gap probability

$\gamma=q^{-M-1}$6

where $\gamma=q^{-M-1}$7. In this setting the q-Racah ensemble is linked to the discrete Painlevé equation

$\gamma=q^{-M-1}$8

and the same analysis yields a Lax pair with an additional involutive symmetry structure (Dzhamay et al., 2019).

A geometrically richer probabilistic realization arises from q-Racah weighted lozenge tilings of a hexagon. For each fixed vertical slice, the white-particle coordinates form a q-Racah ensemble with weight

$\gamma=q^{-M-1}$9

In the scaling regime

$\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$0

the height function concentrates near a deterministic limit shape, and global fluctuations are described by the Gaussian Free Field. The analysis is driven by the asymptotics of the explicitly known q-Racah recurrence coefficients (Duits et al., 2023).

5. Exceptional, multi-indexed, and multivariate extensions

The ordinary q-Racah weight admits several deformations that preserve finite orthogonality but modify the measure. For the exceptional $\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$1 q-Racah family, the support becomes

$\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$2

and the orthogonality weight is

$\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$3

The deforming polynomial $\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$4 is positive on the relevant lattice in the admissible range, and the exceptional family remains a complete orthogonal system even though its lowest-degree member has degree $\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$5, not degree $\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$6 (Odake et al., 2011).

For multi-indexed q-Racah polynomials, the deformed weight is

$\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$7

The same framework also produces dual multi-indexed q-Racah polynomials, whose dual orthogonality measure lives on the degree lattice and has weights proportional to $\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$8. In this dualized form the polynomials are ordinary orthogonal polynomials and Krall-type (Odake, 2018, Odake, 2018).

Multivariate q-Racah measures are equally standard. The Gasper–Rahman multivariate q-Racah polynomials are orthogonal on the simplex

$\sigma(q^{-x})=q^{-x}+\gamma\delta q^{x+1},$9

with respect to a discrete weight $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$0. The same weight appears representation-theoretically as the measure induced by squared overlap coefficients between two orthonormal coupling bases for positive-discrete series representations of $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$1 (Genest et al., 2017).

A different multivariate direction is provided by bivariate Griffiths polynomials of q-Racah type. There the univariate building block is the normalized q-Racah weight

$$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$2

the Tratnik polynomials have a genuine orthogonality weight on the triangular lattice $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$3, $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$4, while the new Griffiths family is generally biorthogonal rather than orthogonal. Accordingly, it does not come with a single canonical probability measure in full generality (Crampe et al., 2024).

6. Related but distinct uses of the term

The phrase “q-Racah probability measure” is not used uniformly. In the q-Racah XX-chain paper, the recurrence coefficients, spectral grid, and Jacobi data are given explicitly, but the paper does not write an orthogonality sum with weights $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$5 and does not explicitly use the phrase “q-Racah probability measure.” The measure is inferred from standard finite q-Racah orthogonality theory. The same paper also separates standard q-Racah from para q-Racah: its second persymmetric regime no longer uses the standard q-Racah measure directly, but passes to para q-Racah by a limiting procedure (Crampe et al., 13 Dec 2025).

An even sharper distinction is provided by the “q-Racah probability distribution”

$$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$6

introduced for Grassmannians over $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$7. This is a finite probability distribution on $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$8 whose probability mass function is expressed by a single q-Racah polynomial and whose cumulative distribution function is expressed by a terminating $$\omega^{\textup{qR}}(x)=\frac{(\alpha q,\beta\delta q,\gamma q,\gamma \delta q;q)_x}{(q,\alpha^{-1}\gamma \delta q, \beta^{-1}\gamma q, \delta q;q )_x}\, \frac{1-\gamma\delta q^{2x+1}}{(\alpha \beta q)^{x}(1-\gamma\delta q)}.$$9. The paper proves positivity by representation-theoretic arguments, and explicitly notes that the q-Racah parameters in the pmf lie outside the usual orthogonality region. This is therefore a probabilistic object encoded by q-Racah polynomials, but not the same as the normalized q-Racah orthogonality weight (Hayashi et al., 2023).

A neighboring but distinct family is the q-generalized para-Racah system. Its orthogonality measure is supported on a q-quadratic bi-lattice rather than on the single q-Racah lattice. Only in the special persymmetric case

$P_n$0

does the bi-lattice collapse to a single lattice, and the polynomials reduce to monic q-Racah polynomials with base $P_n$1. This boundary case clarifies that para-q-Racah and q-Racah measures are related but not identical objects (Lemay et al., 2017).

In summary, the most stable meaning of the q-Racah probability measure is the normalized finite discrete orthogonality weight of the q-Racah polynomials. Around that core lie several compatible reinterpretations—Jacobi spectral measures, orthogonal polynomial ensembles, multivariate overlap measures, and deformed exceptional or multi-indexed weights—as well as distinct probabilistic distributions merely expressed in q-Racah functions.