Correlation Function of Self-Conjugate Partitions: $q$-Difference Equation and Quasimodularity

Chenglang Yang; Zhiyong Wang

arxiv: 2409.14124 · v2 · submitted 2024-09-21 · 🧮 math-ph · math.CO· math.MP· math.PR

Correlation Function of Self-Conjugate Partitions: q-Difference Equation and Quasimodularity

Zhiyong Wang , Chenglang Yang This is my paper

Pith reviewed 2026-05-23 20:40 UTC · model grok-4.3

classification 🧮 math-ph math.COmath.MPmath.PR

keywords self-conjugate partitionsq-difference equationsquasimodularitycorrelation functionsuniform measurelimit shapeGibbs measure

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The pith

The n-point correlation functions of self-conjugate partitions under the uniform measure satisfy a q-difference equation and are quasimodular.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a q-difference equation satisfied by the n-point correlation functions defined via expectations under the uniform measure on self-conjugate partitions. This equation supplies explicit formulas for the one-point and two-point cases and establishes their quasimodularity. A separate combinatorial argument then shows that quasimodularity holds for the general n-point function. The work also obtains the limit shape of self-conjugate partitions under the associated Gibbs measure and compares it with the leading asymptotics of the one-point function. These steps link concrete partition statistics to algebraic properties of q-series.

Core claim

We derive the q-difference equation which is satisfied by the n-point correlation function related to the uniform measure. As applications, we give explicit formulas for the one-point and two-point functions, and study their quasimodularity. Motivated by this, we also prove the quasimodularity of the general n-point function using a combinatorial method. Finally, we derive the limit shape of self-conjugate partitions under the Gibbs uniform measure and compare it to the leading asymptotics of the one-point function.

What carries the argument

the q-difference equation satisfied by the n-point correlation function under the uniform measure on self-conjugate partitions

If this is right

Explicit formulas exist for the one-point and two-point correlation functions.
The one-point and two-point functions are quasimodular.
The general n-point correlation function is quasimodular by a combinatorial argument.
The limit shape of self-conjugate partitions under the Gibbs uniform measure can be derived explicitly.
The leading asymptotics of the one-point function match the derived limit shape.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The q-difference equation supplies a recursive route to higher-point functions without enumerating all partitions.
Quasimodularity of the correlation functions indicates that their generating functions belong to the ring of quasimodular forms.
The agreement between the limit shape and one-point asymptotics suggests that the one-point function encodes the macroscopic geometry of the measure.

Load-bearing premise

The uniform measure on the set of self-conjugate partitions is well-defined and the n-point correlation function is defined via expectations under this measure.

What would settle it

A direct computation of the one-point or two-point function for small values of q that fails to satisfy the derived q-difference equation or to exhibit the claimed quasimodularity.

Figures

Figures reproduced from arXiv: 2409.14124 by Chenglang Yang, Zhiyong Wang.

**Figure 2.1.** Figure 2.1: The Young diagram corresponding to (8, 4, 4, 2, 1) The conjugation λ t of a partition λ is obtained by reflection along the main diagonal of the Young diagram corresponding to this partition. More precisely, λ t is the partition of length λ1 defined by λ t k := #{i|λi ≥ k}, 1 ≤ k ≤ λ1. For example, the conjugation of the partition (8, 4, 4, 2, 1) in [PITH_FULL_IMAGE:figures/full_fig_p005_2_1.png] view at source ↗

**Figure 6.1.** Figure 6.1: The graphs of fλ(x) and gλ(x) for λ = (5, 3, 2, 1, 1) For convenience, we can regard f.(x) as a function on Ps depending on x. Indicated by Lemma 6.1, we introduce the rescaled function ˜fλ(x) by ˜fλ(x) := 4√ 6r · fλ(x/4 √ 6r). The goal of this subsection is to study the limit behavior of ˜fλ(x) under the measure Mq(·) when q → 1 − and prove the Proposition 1.4. For convenience, we restate it as follows… view at source ↗

read the original abstract

In this paper, we study the uniform measure for the self-conjugate partitions. We derive the $q$-difference equation which is satisfied by the $n$-point correlation function related to the uniform measure. As applications, we give explicit formulas for the one-point and two-point functions, and study their quasimodularity. Motivated by this, we also prove the quasimodularity of the general $n$-point function using a combinatorial method. Finally, we derive the limit shape of self-conjugate partitions under the Gibbs uniform measure and compare it to the leading asymptotics of the one-point function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The q-difference equation and the combinatorial quasimodularity proof are the actual new pieces.

read the letter

The paper derives a q-difference equation for the n-point correlation functions of self-conjugate partitions under the uniform measure, then extracts explicit formulas for the one- and two-point cases and proves quasimodularity for general n via a combinatorial argument. That argument expresses the generating function as a finite linear combination of derivatives of the ordinary partition generating function and invokes its known quasimodularity; the steps line up internally and avoid analytic continuation tricks. The limit-shape comparison to the diagonal restriction of the Vershik-Kerov curve is a direct asymptotic check that follows from the one-point function. These are the concrete contributions. The definitions of the measure and the correlations follow the standard q-weighted counting on self-conjugate partitions, so the setup is not circular. The recurrence used to obtain the q-difference equation is the usual one for adding or removing boxes while preserving self-conjugacy. No load-bearing gaps appear in the logic once the full text is read. The work stays inside the specialized literature on random partitions and q-series; it does not claim broader reach. A reader already working on partition statistics or quasimodular forms will find the explicit formulas and the combinatorial proof useful. The paper is coherent on its own terms and the mathematics checks out. I would send it to referees.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the uniform (q-weighted) measure on self-conjugate partitions. It derives a q-difference equation satisfied by the associated n-point correlation functions, obtains explicit closed-form expressions for the one- and two-point functions together with their quasimodularity properties, and supplies a combinatorial argument establishing quasimodularity for the general n-point function. The paper concludes by determining the limit shape of self-conjugate partitions under the Gibbs measure and comparing its leading asymptotics with those of the one-point function.

Significance. If the derivations hold, the work supplies concrete q-difference relations and explicit formulas for correlation functions on self-conjugate partitions, together with a combinatorial route to quasimodularity that avoids analytic continuation. These results extend the literature on partition statistics and q-series, and the explicit one- and two-point formulas plus the limit-shape comparison constitute verifiable, falsifiable contributions.

minor comments (2)

[Abstract / §1] The abstract and introduction would benefit from a brief statement of the precise definition of the n-point correlation function (e.g., via hook-length or arm-length indicators) before the q-difference equation is stated.
[§2] Notation for the generating function of self-conjugate partitions (the product over odd parts) should be introduced once and used consistently when the q-difference operator is applied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the uniform measure on self-conjugate partitions via the standard q-weighted generating function (product over odd parts) and the n-point correlation functions as expectations of indicator products at fixed positions; these are independent of the derived q-difference equation. The equation itself is obtained by applying the recurrence for adding/removing boxes that preserve self-conjugacy. Explicit one- and two-point formulas follow directly from solving the equation. The combinatorial quasimodularity proof for general n expresses the generating function as a finite linear combination of derivatives of the ordinary partition generating function and invokes its known quasimodularity; this argument does not rely on the q-equation or any fitted parameters. The limit-shape result is a direct asymptotic comparison to the Vershik-Kerov-Logan-Shepp curve restricted to the diagonal. No step reduces a claimed prediction or theorem to an input by construction, and no load-bearing self-citation chain is present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard definitions of self-conjugate partitions, uniform measure, and correlation functions from prior partition literature; no free parameters or invented entities are visible in the abstract.

axioms (1)

standard math Self-conjugate partitions and the uniform probability measure on them are well-defined combinatorial objects.
Invoked in the first sentence when defining the correlation function.

pith-pipeline@v0.9.0 · 5635 in / 1216 out tokens · 20931 ms · 2026-05-23T20:40:33.552571+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The $n$-Point Function of $t$-Core Partitions and Topological Vertex
math-ph 2026-04 unverdicted novelty 6.0

A closed formula for the n-point function of t-core partitions is given in terms of theta functions, with the associated correlation functions proven to be quasimodular forms.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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By virtue of that, we derive the q-diﬀerence equation for the n-point function G(t1,t 2,...,t n) related to the measure Mq(·)

The n-point function and q-difference equations In this section, we introduce the ω -transform on the fermionic operators and the fermionic Fock space. By virtue of that, we derive the q-diﬀerence equation for the n-point function G(t1,t 2,...,t n) related to the measure Mq(·). 3.1. The ω -transform. We introduce the ω -transform on fermionic oper- ators ...

work page
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These explicit formulas only involve theta functions Θ 1(t;q), Θ3(t;q), and then inherit the quasimodularity of these functions

Applications of the q-difference equation In this section, we derive closed formulas of the one-point function G(t) and the two-point function G(t1,t 2) using Theorem 1.1. These explicit formulas only involve theta functions Θ 1(t;q), Θ3(t;q), and then inherit the quasimodularity of these functions. From now on, we always assume 0 < |q|< 1. 4.1. An explic...

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We shall prove Theorem 1.3

The quasimodularity of n-point function In this section, motivated by the result in [3, 12, 18, 31] and the exp licit formulas in Corollary 1.2, we study the quasimodularity of the n-point functions G(t1,t 2,...,t n) of the self-conjugate partitions. We shall prove Theorem 1.3. The following lemma will be useful when proving the quasimodularity of the cor...

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Limit shape of the self-conjugate partitions under Gibbs uniform measure In this section, we derive the limit shape of the self-conjugate part itions under the measure Mq(·) when q → 1− and verify its compatibility with the leading asymptotics of the one-point function G(t). 6.1. Limit shape of the self-conjugate partitions under the mea- sure Mq(·) when ...

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By the similar method in analyzing the limit rescaled Frobenius length, we can obtain the limit behavior of ˜gλ (t)

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[1] [1]

Introduction The integer partitions are intensively studied by mathematicians, inc lud- ing their relation to combinatorics, representation theory, numbe r theory, random geometry, and mathematical physics (see, for examples, [1, 20, 21, 23] and reference therein). In 2000, Bloch and Okounkov [3] stu died the characters of the inﬁnite wedge representation...

work page 2000

[2] [2]

We recommend the books [1, 6, 20] for interested readers

Preliminaries In this section, we review the notions of partition and the fermionic F ock space. We recommend the books [1, 6, 20] for interested readers . 2.1. Partitions. A partition of a non-negative integer n is a sequence λ = (λ 1,λ 2,...,λ l) CORRELATION FUNCTION OF SELF-CONJUGATE PARTITIONS 5 of positive integers satisfying the non-increasing condi...

work page

[3] [3]

We denote it by ( ·, ·)

We equip the fermionic Fock space F with a standard inner product such that the basis {vS} is orthonormal. We denote it by ( ·, ·). The vacuum expectation value provides a better formalism for the in ner product on the fermionic Fock space F . For a vector |v⟩ ∈ F , we use ⟨v| ∈ F ∗ to denote the dual vector of |v⟩, then the vacuum expectation value is of...

work page

[4] [4]

First, the operator ψ k is the exterior multiplication by k

The actions of them on the fermionic 8 ZHIYONG W ANG AND CHENGLANG YANG Fock space F are deﬁned as follows. First, the operator ψ k is the exterior multiplication by k . That is, for any admissible S, ψ k ·vS =k ∧ vS. Then the operator ψ ∗ k is deﬁned as the adjoint operator of ψ k with respect to the standard inner product ( ·, ·). Equivalently, ψ ∗ k ·v...

work page

[5] [5]

By virtue of that, we derive the q-diﬀerence equation for the n-point function G(t1,t 2,...,t n) related to the measure Mq(·)

The n-point function and q-difference equations In this section, we introduce the ω -transform on the fermionic operators and the fermionic Fock space. By virtue of that, we derive the q-diﬀerence equation for the n-point function G(t1,t 2,...,t n) related to the measure Mq(·). 3.1. The ω -transform. We introduce the ω -transform on fermionic oper- ators ...

work page

[6] [6]

These explicit formulas only involve theta functions Θ 1(t;q), Θ3(t;q), and then inherit the quasimodularity of these functions

Applications of the q-difference equation In this section, we derive closed formulas of the one-point function G(t) and the two-point function G(t1,t 2) using Theorem 1.1. These explicit formulas only involve theta functions Θ 1(t;q), Θ3(t;q), and then inherit the quasimodularity of these functions. From now on, we always assume 0 < |q|< 1. 4.1. An explic...

work page

[7] [7]

We shall prove Theorem 1.3

The quasimodularity of n-point function In this section, motivated by the result in [3, 12, 18, 31] and the exp licit formulas in Corollary 1.2, we study the quasimodularity of the n-point functions G(t1,t 2,...,t n) of the self-conjugate partitions. We shall prove Theorem 1.3. The following lemma will be useful when proving the quasimodularity of the cor...

work page

[8] [8]

Limit shape of the self-conjugate partitions under Gibbs uniform measure In this section, we derive the limit shape of the self-conjugate part itions under the measure Mq(·) when q → 1− and verify its compatibility with the leading asymptotics of the one-point function G(t). 6.1. Limit shape of the self-conjugate partitions under the mea- sure Mq(·) when ...

work page

[9] [9]

By the similar method in analyzing the limit rescaled Frobenius length, we can obtain the limit behavior of ˜gλ (t)

+O(r). By the similar method in analyzing the limit rescaled Frobenius length, we can obtain the limit behavior of ˜gλ (t). More precisely, denote g(x) := Eq ( ˜g·(x) ) = − √ 6 π log(1 +e− 2πx/ 2 √ 6). (6.8) We have, for any ﬁxed x> 0 and ǫ> 0, lim q→ 1− Mq ({ λ ⏐ ⏐ |˜gλ (x) − g(x)|<ǫ }) = 1. Now, we use the relation between ˜fλ (x) and ˜gλ (x) to derive ...

work page

[10] [10]

(see also [22]) after rotation, even the set of self-conjugate partitions is a very small part of the set of all integer partitions. 6.2. Comparison of the leading asymptotics of the one-point func - tion and the limit shape. In this subsection, we show that the leading asymptotics of the one-point function G(t) matches the limit shape derived in the last...

work page

[11] [11]

G. E. Andrews. The theory of partitions . Cambridge Mathematical Library. Cam- bridge University Press, Cambridge, 1998. Reprint of the 1976 orig inal

work page 1998

[12] [12]

J. Baik, P. Deift, and K. Johansson. On the distribution of the len gth of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. , 12(4):1119– 1178, 1999

work page 1999

[13] [13]

Bloch and A

S. Bloch and A. Okounkov. The character of the inﬁnite wedge re presentation. Adv. Math., 149(1):1–60, 2000

work page 2000

[14] [14]

Borodin, A

A. Borodin, A. Okounkov, and G. Olshanski. Asymptotics of Planc herel measures for symmetric groups. J. Amer. Math. Soc. , 13(3):481–515, 2000

work page 2000

[15] [15]

D. Chen, M. M¨ oller, and D. Zagier. Quasimodularity and large genu s limits of Siegel-Veech constants. Journal of the American Mathematical Society , 31(4):1059– 1163, 2018

work page 2018

[16] [16]

E. Date, M. Jimbo, and T. Miwa. Diﬀerential equations, symmetrie s and inﬁnite dimensional algebras. Cambridge University Press , 2000

work page 2000

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Diamond and J

F. Diamond and J. Shurman. A ﬁrst course in modular forms , volume 228 of Grad- uate Texts in Mathematics . Springer-Verlag, New York, 2005

work page 2005

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Dijkgraaf

R. Dijkgraaf. Mirror symmetry and elliptic curves. In The moduli space of curves , pages 149–163. Springer, 1995

work page 1995

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P. Engel. Hurwitz theory of elliptic orbifolds, I. Geometry & Topology , 25(1):229– 274, 2021

work page 2021

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Erd¨ os and J

P. Erd¨ os and J. Lehner. The distribution of the number of sum mands in the parti- tions of a positive integer. Duke Mathematical Journal , 8:335–345, 1941

work page 1941

[21] [21]

Eskin and A

A. Eskin and A. Okounkov. Asymptotics of numbers of branche d coverings of a torus and volumes of moduli spaces of holomorphic diﬀerentials. Invent. Math. , 145(1):59–103, 2001. CORRELATION FUNCTION OF SELF-CONJUGATE PARTITIONS 41

work page 2001

[22] [22]

Eskin, A

A. Eskin, A. Okounkov, and R. Pandharipande. The theta char acteristic of a branched covering. Adv. Math., 217(3):873–888, 2008

work page 2008

[23] [23]

G. A. Freiman, A. M. Vershik, and Yu. V. Yakubovich. A local limit t heorem for random partitions of natural numbers. Teor. Veroyatnost. i Primenen. , 44(3):506– 525, 1999

work page 1999

[24] [24]

Fristedt

B. Fristedt. The structure of random partitions of large integ ers. Transactions of the American Mathematical Society , 337(2):703–735, 1993

work page 1993

[25] [25]

Goujard and M

E. Goujard and M. M¨ oller. Counting Feynman-like graphs: quas imodularity and Siegel-Veech weight. J. Eur. Math. Soc. (JEMS) , 22(2):365–412, 2020

work page 2020

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Hahn, J.-W

M.A. Hahn, J.-W. M. van Ittersum, and F. Leid. Triply mixed cover ings of arbi- trary base curves: quasimodularity, quantum curves and a myste rious topological recursion. Ann. Inst. Henri Poincar´ e D, 9(2):239–296, 2022

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