arXiv subjects

Zhengye Zhou

Publications and source records attributed to Zhengye Zhou.

13 recordsLinked to original sources

arXiv2026

From the Airy line ensemble to the Gaussian free field

We study the global fluctuations of the height function associated with the Airy line ensemble. Using its determinantal structure and a steepest-descent analysis of the extended Airy kernel, we prove that, after a suitable rescaling, the centered height function converges to an explicit pullback of the Gaussian free field. The convergence holds in the sense of joint moments of linear statistics against compactly supported continuous test functions.

math.PR↗

arXiv2026

Orthogonal Dualities of Dynamic Stochastic Higher Spin Vertex Models, using the Drinfeld Twister

We introduce a new algebraic method to construct duality functions for integrable dynamic models. This method will be implemented on dynamic stochastic higher spin vertex models, where we prove that the resulting duality functions between the dynamic stochastic higher spin vertex models and non-dynamic stochastic higher spin vertex models are the ${}_3 φ_2$ functions. A degeneration of these duality functions is dual $q$-Krawtchouk polynomials, which agree with the orthogonal polynomial dualities of Groenevelt--Wagenaar arXiv:2306.12318 between dynamic ASEP and ASEP. The method relies on the universal twister of $U_q(\mathfrak{sl}_2)$, regarded as a quasi-triangular quasi-Hopf algebra. Since the algebraic construction is formulated in a general setting, it is expected to produce duality functions for many other dynamic integrable models as well.

math.PR↗

arXiv2026

Curve separation in supercritical half-space last passage percolation

We study line ensembles arising naturally in symmetrized/half-space geometric last passage percolation (LPP) on the $N \times N$ square. The weights of the model are geometrically distributed with parameter $q^2$ off the diagonal and $cq$ on the diagonal, where $q \in (0,1)$ and $c \in [0, q^{-1})$. In the supercritical regime $c > 1$, we show that the ensembles undergo a phase transition: the top curve separates from the rest and converges to a Brownian motion under $N^{1/2}$ fluctuations and $N$ spatial scaling, while the remaining curves converge to the Airy line ensemble under $N^{1/3}$ fluctuations and $N^{2/3}$ spatial scaling. Our analysis relies on a distributional identity between half-space LPP and the Pfaffian Schur process. The latter exhibits two key structures: (1) a Pfaffian point process, which we use to establish finite-dimensional convergence of the ensembles, and (2) a Gibbsian line ensemble, which we use to extend convergence uniformly over compact sets.

math.PR↗

arXiv2026

Uniform convergence of Pfaffian point process to the Airy line ensemble

We consider a family of Pfaffian Schur processes whose first coordinate marginal relates to the half--space geometric last passage percolation. We show that the line ensembles corresponding to the Pfaffian Schur processes with geometric weights converge uniformly over compact sets to the Airy line ensemble. By detailed asymptotic analysis of the kernels, we can verify the conditions for the finite dimensional weak convergence introduced in arXiv:2408.08445. By utilizing the tightness criteria of the line ensembles established in arXiv:2410.23899, we can further improve the finite dimensional convergence to the uniform convergence over compact sets. Moreover, using the same methodology we also show that sequences of spiked Pfaffian Schur processes converge uniformly over compact sets to the Airy wanderer line ensembles constructed in arXiv:2408.08445.

math.PR↗

arXiv2024

Asymptotics of dynamic ASEP using duality

Using a recently developed method for proving asymptotics via orthogonal polynomial duality arXiv:2305.17602, we prove that the dynamic ASEP introduced in arXiv:1701.05239 has asymptotics which are either distributed as the Tracy--Widom $F_2,$ or are almost surely bounded. Using a different duality, we also provide contour integrals formulas for multi--species ASEP, which generalize results for the single--species ASEP.

math.PR↗

arXiv2024

Asymptotics of two-point correlations in the multi-species q-TAZRP

A previous paper by the authors found explicit contour integral formulas for certain joint moments of the multi-species q-TAZRP (totally asymmetric zero range process), using algebraic methods. These contour integral formulas have a "pseudo-factorized" form which makes asymptotic analysis simpler. In this brief note, we use those contour integral formulas to find the asymptotics of the two-point correlations. As expected, the term arising from the "shift-invariance" makes a non-trivial asymptotic contribution.

math.PR↗

arXiv2024

Orthogonal polynomial duality and unitary symmetries of multi--species ASEP$(q,\boldsymbolθ)$ and higher--spin vertex models via $^*$--bialgebra structure of higher rank quantum groups

We propose a novel, general method to produce orthogonal polynomial dualities from the $^*$--bialgebra structure of Drinfeld--Jimbo quantum groups. The $^*$--structure allows for the construction of certain \textit{unitary} symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group $\mathcal{U}_q(\mathfrak{gl}_{n+1})$, the result is a nested multivariate $q$--Krawtchouk duality for the $n$--species ASEP$(q,\boldsymbolθ)$. The method also applies to other quantized simple Lie algebras and to stochastic vertex models. As a probabilistic application of the duality relation found, we provide the explicit formula of the $q-$shifted factorial moments (namely the $q$-analogue of the Pochhammer symbol) for the two--species $q$--TAZRP (totally asymmetric zero range process).

math.PR↗

arXiv2024

Orthogonal polynomial duality of a two-species asymmetric exclusion process

We examine type D ASEP, a two--species interacting particle system which generalizes the usual asymmetric simple exclusion process. For certain cases of type D ASEP, the process does not give priority for one species over another, even though there is nontrivial interaction between the two species. For those specific cases, we prove that the type D ASEP is self--dual with respect to an independent product of $q$--Krawtchouk polynomials. The type D ASEP was originally constructed in arXiv:2011.13473, using the type D quantum groups $\mathcal{U}_q(\mathfrak{so}_6)$ and $\mathcal{U}_q(\mathfrak{so}_8)$. That paper claimed that certain states needed to be "discarded'' in order to ensure non--negativity. Here, we also provide a more efficient argument for the same claim.

math.PR↗

arXiv2022

Three-dimensional Gaussian fluctuations of non-commutative random surface growth with a reflecting wall

We consider the multi-time correlation and covariance structure of a random surface growth with a wall introduced in arXiv:0904.2607. It is shown that the correlation functions associated with the model along space-like paths have determinantal structure, which yields the convergence of height fluctuations to that of a Gaussian free field. We also construct a continuous-time non-commutative random walk on $U(\mathfrak{so}_{N+1})$, which matches the random surface growth when restricting to the Gelfand-Tsetlin subalgebra of $U(\mathfrak{so}_{N+1})$. As an application, we prove the convergence of moments to an explicit Gaussian free field and get the covariance functions of the associated random point process along both the space-like paths and time-like paths. In particular, it does not match the three-dimensional Gaussian field from spectra of overlapping stochastic Wishart matrices in arXiv:2112.13728 even along the space-like paths.

math.PR↗

arXiv2021

Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices

In arXiv:1410.7268v3, the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic evolution, the fluctuations asymptotically converge to a three-dimensional Gaussian field, which has an explicit contour integral formula. This is analogous to the result of arXiv:1011.3544 for stochastic Wigner matrices.

math.PR↗

arXiv2021

Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP$(2j)$ and Multi-Species IRW

We obtain orthogonal polynomial self-duality functions for multi-species version of the symmetric exclusion process (SEP$(2j)$) and the independent random walker process (IRW) on a finite undirected graph. In each process, we have $n>1$ species of particles. In addition, we allow up to $2j$ particles to occupy each site in the multi-species SEP$(2j)$. The duality functions for the multi-species SEP$(2j)$ and the multi-species IRW come from unitary intertwiners between different $*$-representations of the special linear Lie algebra $\mathfrak{sl}_{n+1}$ and the Heisenberg Lie algebra $\mathfrak{h}_n$, respectively. The analysis leads to multivariate Krawtchouk polynomials as orthogonal duality functions for the multi-species SEP$(2j)$ and homogeneous products of Charlier polynomials as orthogonal duality functions for the multi-species IRW.

math.PR↗

arXiv2020

Interacting particle systems with type D symmetry and duality

We construct a two-class asymmetric interacting particle system with $U_q(so_6)$ or $U_q(so_8)$ symmetry, in which up to two particles may occupy a site if the two particles have different class. The particles exhibit a drift, but there is no preference given between first-class and second-class particles. The quantum group symmetry leads to reversible measures and a self-duality for the particle system. Additionally, a new method is developed to construct a symmetric interacting particle system from the Casimir element of $so_{2n}$.

math.PR↗

arXiv2020

Hydrodynamic limit for a $d$-dimensional open symmetric exclusion process

In this paper we focus on the open symmetric exclusion process with parameter $m$ (open SEP($m/2$)), which allows $m$ particles each site and has an open boundary. We generalize the result about hydrodynamic limit for the open SEP$(m/2)$ that was originally raised in Theorem 4.12 of arXiv:1908.02359. We prove that the hydrodynamic limit of the density profile for a $d-$dimensional open SEP$(m/2)$ solves the $(d+1)-$dimensional heat equation with certain initial condition and boundary condition.

math.PR↗