arXiv subjects

Zongrui Yang

Publications and source records attributed to Zongrui Yang.

17 recordsLinked to original sources

arXiv2026

A note on last passage percolation and Schur processes

In this note we provide a short proof of the distributional equality between last passage percolation with geometric weights along a general down-right path and Schur processes. We do this in both the full-space and half-space settings, and for general parameters. The main inputs for our arguments are generalizations of the Robinson-Schensted-Knuth correspondence and Greene's theorem due to Krattenthaler, which are based on Fomin's growth diagrams.

math.PR↗

arXiv2026

Stationary measures for higher spin vertex models on a strip

We introduce a higher spin vertex model on a strip with fused vertex weights. This model can be regarded as a generalization of both the unfused six-vertex model on a strip arXiv:2212.09111 and an 'integrable two-step Floquet dynamics' model introduced in arXiv:1711.08884. We solve for the stationary measure using a fused version of the matrix product ansatz and then characterize it in terms of the Askey-Wilson process. Using this characterization, we obtain the limits of the mean density along an arbitrary down-right path. It turns out that all these models share a common phase diagram, which, after an appropriate mapping, matches the phase diagram of open ASEP. This provides evidence for the universality of this phase diagram.

math-ph↗

arXiv2026

Invariant measures for half-space geometric LPP: classification and the one force--one solution principle

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class that has boundary effects and an unbounded domain. A description of a class of invariant measures was previously given in a work of Barraquand and Corwin, where it was conjectured that these should comprise all extremal invariant measures. To complete the classification, we prove a one force--one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at $\infty$, the recentered solution at time $0$ converges to a process which is distributed as the associated invariant measure with the specified slope. This limiting process is called the Busemann process, the first of its kind constructed for a half-space model. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP, recently constructed by Dauvergne and Zhang. There, it was conjectured that the constructed family of jointly invariant measures comprises all extremal jointly invariant measures; our analysis also confirms this conjecture. When the model has a strong (attractive) boundary, the collection of slopes for the invariant measures has a discontinuity, which does not arise in the full-space case. To handle this difficulty, we combine the control of the directions of semi-infinite geodesics with techniques from the theory of half-space Gibbsian line ensembles. Along the way, we classify the set of directions of semi-infinite geodesics for half-space geometric LPP, confirming a recent conjecture of Dauvergne and Zhang.

math.PR↗

arXiv2026

Pinning in non-critical half-space geometric last passage percolation

We study a symmetrized (half-space) version of geometric last passage percolation with a boundary parameter $c$ that interpolates between subcritical, critical, and supercritical behavior. This model gives rise to a family of interlacing random curves, or a line ensemble, which encode both the usual last passage time and its higher-rank analogues. Although these ensembles are understood in most space-time regions, their behavior near the diagonal -- where the boundary effects are strongest -- has remained unclear outside the critical regime. We determine the universal scaling limits of the line ensemble in this near-diagonal region for both subcritical ($c < 1$) and supercritical ($c > 1$) phases. In the subcritical case, after appropriate centering and scaling, the entire line ensemble converges to the pinned half-space Airy line ensemble, a universal Brownian Gibbsian object recently constructed as a canonical limit for half-space models in the KPZ universality class in arXiv:2601.04546. In the supercritical case, we prove an analogous convergence together with a curve-separation phenomenon: the lower curves converge to the same pinned half-space Airy limit, while the top curve decouples and converges to Brownian motion. These results essentially complete the asymptotic description of half-space geometric last passage percolation and provide a new rigorous instance of the pinned half-space Airy line ensemble as a universal scaling limit.

math.PR↗

arXiv2026

Half-space Airy line ensembles

We construct a one-parameter family of infinite line ensembles on $[0, \infty)$ that are natural half-space analogues of the Airy line ensemble. Away from the origin these ensembles are locally described by avoiding Brownian bridges, and near the origin they are described by a sequence of avoiding reverse Brownian motions with alternating drifts, that depend on the parameter of the model. In addition, the restrictions of our ensembles to finitely many vertical lines form Pfaffian point processes with the crossover kernels obtained by Baik-Barraquand-Corwin-Suidan (Ann. Probab., 46(6), 3015-3089, 2018)

math.PR↗

arXiv2026

The pinned half-space Airy line ensemble

Half-space models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit rich boundary phenomena that alter the asymptotic behavior familiar from their full-space counterparts. A distinguishing feature of these systems is the presence of a boundary parameter that governs a transition between subcritical, critical, and supercritical regimes, characterized by different scaling exponents and fluctuation statistics. In this paper we construct the pinned half-space Airy line ensemble $\mathcal{A}^{\mathrm{hs}; \infty}$ on $[0,\infty)$ -- a natural half-space analogue of the Airy line ensemble -- expected to arise as the universal scaling limit of supercritical half-space KPZ models. The ensemble $\mathcal{A}^{\mathrm{hs}; \infty}$ is obtained as the weak limit of the critical half-space Airy line ensembles $\mathcal{A}^{\mathrm{hs}; \varpi}$ introduced in arXiv:2505.01798 as the boundary parameter $\varpi$ tends to infinity. We show that $\mathcal{A}^{\mathrm{hs}; \infty}$ has a Pfaffian point process structure with an explicit correlation kernel and that, after a parabolic shift, it satisfies a one-sided Brownian Gibbs property described by pairwise pinned Brownian motions. Far from the origin, $\mathcal{A}^{\mathrm{hs}; \infty}$ converges to the standard Airy line ensemble, while at the origin its distribution coincides with that of the ordered eigenvalues (with doubled multiplicity) of the stochastic Airy operator with $β= 4$.

math.PR↗

arXiv2025

Limits of open ASEP stationary measures near a boundary

Consider the stationary measure of open asymmetric simple exclusion process (ASEP) on the lattice $\{1,\dots,n\}$. Taking $n$ to infinity while fixing the jump rates, this measure converges to a measure on the semi-infinite lattice. In the high and low density phases, we characterize the limiting measure and provide bounds on the convergence rates in total variation distance. Our approach involves bounding the total variation distance using generating functions, which are further estimated through a subtle analysis of the atom masses of Askey-Wilson signed measures.

math.PR↗

arXiv2025

From asymmetric simple exclusion processes with open boundaries to stationary measures of open KPZ fixed point: the shock region

We continue the investigation of limit fluctuations of stationary measures of the asymmetric simple exclusion processes with open boundaries (open ASEP), complementing the recent result by Bryc et al. (2023). It was shown therein that in the fan region of the phase diagram, an appropriate scaling limit of the height function of open ASEP converges in distribution to a stochastic process introduced by Barraquand and Le Doussal (2022), known as the stationary measure of the (conjectural) open KPZ fixed point. In this paper, we establish the corresponding convergence in the shock region. Our proof is based on the integral representation of the matrix product ansatz in terms of Askey-Wilson signed measures introduced by Wang et al. (2024). The analysis of the asymptotic behavior is more delicate here than in the fan region. In particular, our proof of the duality formula for the stationary measures of the open KPZ fixed point in the shock region is different from the approach by Bryc et al. (2023) taken in the fan region: ours relies crucially on a recent result by Bryc and Zatitskii (2024).

math.PR↗

arXiv2024

Large deviation principle for the stationary measures of open asymmetric simple exclusion processes

We consider the stationary measure of the asymmetric simple exclusion process (ASEP) on a finite interval in $\mathbb{Z}$ with open boundaries. Fixing all the jump rates and letting the system size approach infinity, the height profile of such a sequence of stationary measures satisfies a large deviation principle (LDP), whose rate function was predicted in the physics work arXiv:cond-mat/0205353. In this paper, we provide the first rigorous proof of the large deviation principle in the "fan region" part of the phase diagram. Our proof relies on two key ingredients: a two-layer expression of the stationary measure of open ASEP, arising from the Enaud-Derrida representation arXiv:cond-mat/0307023 of the matrix product ansatz, and the large deviation principle of the open totally asymmetric simple exclusion process (TASEP) recently established in arXiv:2403.03275.

math.PR↗

arXiv2024

The open ASEP with light particles

We consider the stationary measure of the open asymmetric simple exclusion process (ASEP) with light particles. We prove several results on the asymptotic locations of the light particles under the stationary measure. Moreover, in the fan region of the high and low density phases, we study the mixing times for open ASEP with light particles. Our approach involves the matrix product ansatz, Askey-Wilson signed measures, moderate deviation results for second class particles, as well as various coupling arguments. Along the way we obtain the particle densities at single sites for the standard open ASEP without light particles, which are of independent interest.

math.PR↗

arXiv2024

Stationary measures for integrable polymers on a strip

We prove that the stationary measures for the free-energy increment process for the geometric last passage percolation (LPP) and log-gamma polymer model on a diagonal strip is given by a marginal of a two-layer Gibbs measure with a simple and explicit description. This result is shown subject to certain restrictions on the parameters controlling the weights on the boundary of the strip. However, from this description and an analytic continuation argument we are able to access the stationary measure for all boundary parameters. Taking an intermediate disorder limit of the log-gamma polymer stationary measure in a strip we readily recover (modulo convergence of the polymer to the open KPZ equation, Conjecture 4.2) the conjectural description from arXiv:2105.15178 of the open KPZ stationary measure for all choices of boundary parameters $u,v\in \mathbb{R}$ (thus going beyond the restriction $u+v\geq 0$ from arXiv:2103.12253).

math.PR↗

arXiv2024

Stationary measure for six-vertex model on a strip

We study the stochastic six-vertex model on a strip $$\left\{(x,y)\in\mathbb{Z}^2: 0\leq y\leq x\leq y+N\right\}$$ with two open boundaries. We develop a `matrix product ansatz' method to solve for its stationary measure, based on the compatibility of three types of local moves of any down-right path. The stationary measure on a horizontal path turns out to be a tilting of the stationary measure of the asymmetric simple exclusion process (ASEP) with two open boundaries. Similar to open ASEP, the statistics of this tilted stationary measure as the number of sites $N\rightarrow\infty$ (with the bulk and boundary parameters fixed) also exhibit a phase diagram, which is a tilting of the phase diagram of open ASEP. We study the limit of mean particle density as an example.

math.PR↗

arXiv2024

Askey-Wilson signed measures and open ASEP in the shock region

We introduce a family of multi-dimensional Askey-Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey-Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.

math.PR↗

arXiv2022

Constructing the LG/CY isomorphism between $tt^*$ geometries

For a nondegenerate homogeneous polynomial $f\in\mathbb{C}[z_0, \dots, z_{n+1}]$ with degree $n+2$, we can obtain a $tt^*$ structure from the Landau-Ginzburg model $(\C^{n+2}, f)$ and a (new) $tt^*$ structure on the Calabi-Yau hypersurface defined by the zero locus of $f$ in $\C P^{n+1}$. We can prove that the big residue map considered by Steenbrink gives an isomorphism between the two $tt^*$ structures. We also build the correspondence for non-Calabi-Yau cases, and it turns out that only partial structure can be preserved. As an application, we show that the $tt^*$ geometry structure of Landau-Ginzburg model on relavant deformation space uniquely determines the $tt^*$ geometry structure on Calabi-Yau side. This explains the folklore conclusion in physical literature. This result is based on our early work \cite{FLY}.

math.AG↗

arXiv2022

Twisted Conormal Bundles and Canonical Relations

I study a special type of canonical relations given by twisted conormal bundles, construct a "subcategory" of the symplectic "category" out of these canonical relations and quantize them into semi-classical Fourier integral operators. Furthermore, I give a description of the intrinsic line bundle of symbols of these operators and describe how the symbols compose when the operators compose.

math.SG↗

arXiv2022

Gamma conjecture I for blowing up $\mathbb{P}^n$ along $\mathbb{P}^r$

Consider the Fano manifold $X$ formed by blowing up $\mathbb{P}^n$ along its linear subspace $\mathbb{P}^r$, we check the conifold conditions [3, 1] for its mirror Laurent polynomial $f$, which can imply that $X$ satisfies both Conjecture $\mathcal{O}$ and Gamma conjecture I by Galkin-Golyshev-Iritani [2].

math.AG↗

arXiv2020

LG/CY correspondence between $tt^*$ geometries

The concept of $tt^*$ geometric structure was introduced by physicists (see \cite{CV1, BCOV} and references therein) , and then studied firstly in mathematics by C. Hertling \cite{Het1}. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of a two dimensional topological field theory. In this paper, we propose the LG/CY correspondence conjecture for $tt^*$ geometry and obtain the following result. Let $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ be a nondegenerate homogeneous polynomialof degree $n+2$, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity $(\mathbb{C}^{n+2}, f)$, both can be written as a $tt^*$ structure. We proved that there exists a $tt^*$ substructure on Landau-Ginzburg side, which should correspond to the $tt^*$ structure from variation of Hodge structures in Calabi-Yau side. We build the isomorphism of almost all structures in $tt^*$ geometries between these two models except the isomorphism between real structures.

math.AG↗