Pfaffian Schur Process

• Pfaffian Schur process is a Schur-type integrable probabilistic model defined by Pfaffian kernels, contrasting traditional determinantal structures.
• It leverages free- and neutral-fermion formalisms along with Pfaffian identities to derive explicit multipoint correlation formulas.
• Its framework unifies algebraic and scaling techniques, connecting strict partitions, half-space integrable models, and universal KPZ asymptotics.

The Pfaffian Schur process is a Schur-type integrable probabilistic object whose correlation functions are Pfaffians of skew-symmetric matrix kernels rather than determinants. In the Borodin–Rains formulation, it is a probability measure on chains of ordinary partitions with a boundary term built from $\tau_\lambda$ and products of skew Schur functions, and its associated point configuration on shifted row coordinates is Pfaffian (Baik et al., 2016). Closely related BKP models on strict partitions, such as the shifted Schur measure with Schur $Q/P$-weights, are Pfaffian Schur-type processes in the neutral-fermion sense and provide the strict-partition analogue of the determinantal Schur process (Wang et al., 2018). Across these formulations, the subject sits at the intersection of symmetric functions, free-fermion or neutral-fermion formalisms, Pfaffian identities, and half-space integrable probability.

1. Algebraic foundations

A central algebraic input is that Schur $Q$-functions admit Pfaffian representations, and sums over strict partitions can therefore be collapsed to single Pfaffians. In Okada’s formulation, for a strict partition $\lambda$, one has a Pfaffian expression

$$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$

where the entries of $M(\lambda;x)$ are built from the functions $q_r(x)$ defined by

$$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$

The same paper establishes Pfaffian analogues of Cauchy–Binet and of the Ishikawa–Wakayama minor-summation formula, including

$$\sum_{\substack{I\subset [m]\ |I|\text{ even}}} \operatorname{Pf}(S_I)\det(T_I)=\operatorname{Pf}(T^TST),$$

and uses these identities to give Pfaffian proofs of Schur $Q$-function identities such as

$Q/P$0

This is the algebraic backbone behind Pfaffian Schur processes: sums over index sets or strict partitions become single Pfaffians of kernels built from the underlying data (Okada, 2017).

The same BKP structure appears in the neutral-fermion realization of Schur $Q/P$1-functions. In the BKP hierarchy, neutral fermions $Q/P$2 satisfy

$Q/P$3

and the shifted Schur measure on strict partitions is expressed in terms of Schur $Q/P$4-functions. The resulting correlation functions are Pfaffians of kernels given by neutral-fermion two-point functions, making the BKP hierarchy the Pfaffian counterpart of the KP hierarchy behind the ordinary Schur process (Wang et al., 2018).

A useful consequence is that Pfaffian Schur processes are not defined by a single formula alone; rather, they are a class of Schur-type measures for which Schur, skew Schur, or Schur $Q/P$5-weights combine with Pfaffian summation identities and Pfaffian Wick rules.

2. Process-level definitions

In one standard formulation, the Pfaffian Schur process is a measure on chains of ordinary partitions

$Q/P$6

with weight

$Q/P$7

where $Q/P$8, and the partition function is

$Q/P$9

The associated point process is formed from the shifted row coordinates

$Q$0

and its correlation functions are Pfaffians of a $Q$1 matrix kernel with contour-integral entries $Q$2 (Ghosal, 2017).

A closely related free-boundary formulation considers sequences

$Q$3

with weight

$Q$4

When one boundary is free and the other fixed, this framework recovers the Pfaffian Schur process; when both boundaries are free, the unshifted process is not Pfaffian in general, but a random vertically shifted version is (Betea et al., 2017).

Formulation	State space	Characteristic weight
Borodin–Rains Pfaffian Schur process	Chains of ordinary partitions	$Q$5 times skew Schur factors
One-free-boundary Schur process	Chains with one free boundary	Free-boundary Schur weight $Q$6
Shifted Schur / BKP Pfaffian Schur-type model	Strict partitions	$Q$7 or $Q$8

In the literature, these descriptions are complementary rather than contradictory. The ordinary-partition version is the Borodin–Rains process itself, while strict-partition BKP models are explicitly described as Pfaffian Schur-type processes (Aida et al., 15 May 2026).

3. Fermions, kernels, and derivation methods

The free-fermion formalism provides one derivation of Pfaffianity. In the free-boundary Schur process, the vertex operators $Q$9 and free boundary states

$\lambda$0

satisfy reflection relations encoding Schur–Littlewood identities. In the one-free-boundary case, a generalized Wick lemma yields Pfaffian formulas directly for multipoint correlations, leading to an explicit double-contour kernel with entries $\lambda$1 (Betea et al., 2017).

In the BKP strict-partition setting, the neutral-fermion field

$\lambda$2

and the wave function

$\lambda$3

produce the scalar kernel

$\lambda$4

and the correlation function of the shifted Schur measure is the Pfaffian of the matrix built from $\lambda$5. This realizes the shifted Schur measure as a Pfaffian point process arising from BKP free fermions (Aida et al., 15 May 2026).

An alternative derivation uses Macdonald difference operators. For the Pfaffian Schur process on ordinary partitions, Macdonald $\lambda$6-difference operators act diagonally on Schur functions and convert correlation observables into contour integrals. Ghosal’s derivation reproduces the Borodin–Rains kernel without using the Pfaffian Eynard–Mehta theorem, yielding the same $\lambda$7 contour kernel $\lambda$8 for the point process $\lambda$9 (Ghosal, 2017).

At a more abstract level, a self-dual quasi-free state on the doubled one-particle space $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$0 with covariance $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$1 produces a Pfaffian point process with kernel

$$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$2

This operator-theoretic formalism places Pfaffian Schur-type ensembles inside the general theory of Pfaffian point processes from self-dual CAR algebras (Koshida, 2020).

4. Half-space realizations and solvable models

Pfaffian Schur processes are central in half-space integrable probability. In half-space last passage percolation, the geometric model is coupled exactly to a Pfaffian Schur process: for a suitable space-like path, the vector of passage times $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$3 has the same law as the vector of first rows $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$4 of a Pfaffian Schur process with specializations determined by the path data (Baik et al., 2016). After the exponential limit, one obtains Fredholm Pfaffian formulas for half-space exponential LPP. On the diagonal, fluctuations are GSE Tracy–Widom for $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$5, GOE Tracy–Widom for $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$6, and Gaussian for $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$7; away from the diagonal they are GUE Tracy–Widom (Baik et al., 2016).

A closely related half-space object is the free boundary Schur measure

$$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$8

normalized by a skew Littlewood identity. After theta shift-mixing, it becomes a Pfaffian point process with $$Q_\lambda(x_1,\dots,x_n)=\operatorname{Pf}(M(\lambda;x)),$$9 kernel

$M(\lambda;x)$0

and the rightmost point is governed by a Fredholm Pfaffian (Imamura et al., 2022).

This free-boundary Schur measure is the half-space Schur object behind several KPZ-class models. It underlies Pfaffian formulas for the half-space $M(\lambda;x)$1-PushTASEP with particle creation, the half-space Log Gamma polymer, and the half-line KPZ equation. For the half-space stochastic heat equation with Robin parameter $M(\lambda;x)$2, the Laplace transform of the solution at the origin is

$M(\lambda;x)$3

with

$M(\lambda;x)$4

In this sense, the free-boundary Schur measure is the positive-temperature half-space Pfaffian Schur structure governing one-point KPZ observables (Imamura et al., 2022).

5. Scaling limits and universality

Several scaling theories clarify how Pfaffian Schur processes behave asymptotically. For the specialized geometric Pfaffian Schur processes related to half-space geometric LPP, the corresponding line ensembles converge uniformly over compact sets to the Airy line ensemble. More precisely, the rescaled line ensembles $M(\lambda;x)$5 converge weakly in $M(\lambda;x)$6 to the parabolic Airy line ensemble in the homogeneous regime, and to Airy wanderer line ensembles in spiked regimes; the kernel asymptotics show that the Pfaffian kernel degenerates so that only the extended Airy-type $M(\lambda;x)$7-block survives in the limit (Zhou, 11 Mar 2025).

A distinct BKP phenomenon appears for the shifted Schur measure. Under an appropriate multicritical scaling limit and specific conditions on the continuous parameters, the edge scaling limit of the correlation function converges to a determinant of the higher-order Airy kernel. The kernel entries satisfy

$M(\lambda;x)$8

for the diagonal blocks, while the off-diagonal block converges to $M(\lambda;x)$9. The Pfaffian matrix therefore becomes block-off-diagonal, and the Pfaffian reduces to a determinant. The paper states this as a rigorous Pfaffian-to-determinantal transition in the scaling limit (Aida et al., 15 May 2026).

In the bulk of the same shifted Schur model, the Pfaffian structure also collapses to a determinantal sine-kernel limit: $q_r(x)$0 This suggests that, in these BKP models, Pfaffian discrete structure and determinantal continuum universality can coexist without contradiction (Aida et al., 15 May 2026).

Half-space KPZ asymptotics provide another universality picture. For free-boundary Schur measures and their scaling limits, the half-line KPZ equation exhibits the Baik–Rains phase transition: as the boundary parameter crosses $q_r(x)$1, fluctuations pass from GSE Tracy–Widom through a crossover distribution to Gaussian behavior (Imamura et al., 2022).

6. Variants, extensions, and current directions

Recent work has extended the Pfaffian Schur framework in several directions. A shifted $q_r(x)$2-Schur weight on strict partitions,

$q_r(x)$3

is defined by the modified odd Greaves–Jing–Zhu operator and satisfies

$q_r(x)$4

Its normalization is

$q_r(x)$5

and the associated one-time process has a Pfaffian correlation kernel and a Fredholm Pfaffian for the largest part. At $q_r(x)$6, the virtual alphabet becomes the positive alphabet $q_r(x)$7, giving a genuine probability measure (Lee, 2 Jul 2026).

At the same specialization $q_r(x)$8, inserting an intermediate strict partition produces a two-color lift on pairs $q_r(x)$9 with weight

$$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$0

This model has normalization $$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$1, both marginals are shifted Schur measures, the transition kernel

$$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$2

satisfies the semigroup property $$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$3, and the model is realized as a two-time shifted Schur process with a Pfaffian correlation kernel in Vuletić’s convention (Lee, 2 Jul 2026).

On the algebraic side, ninth-variation skew $$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$4-functions admit a Pfaffian outside-decomposition identity extending Hamel’s Pfaffian formula. This yields Pfaffian identities generalizing those of Józefiak–Pragacz, Nimmo, and Okada, and suggests a wide class of content-dependent $$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$5-weighted Pfaffian Schur-type processes on strict partitions (Foley et al., 2020).

Taken together, these developments indicate that “Pfaffian Schur process” names a robust integrable paradigm rather than a single model: a family of Schur- or Schur $$\sum_{r\ge 0} q_r(x) z^r = \prod_{i=1}^n \frac{1+x_i z}{1-x_i z}.$$6-based measures with Pfaffian correlation structure, realized through free or neutral fermions, contour-integral kernels, and Pfaffian identities, and connected to half-space KPZ models, BKP hierarchies, and strict-partition asymptotics (Betea et al., 2017).