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Yizao Wang

Publications and source records attributed to Yizao Wang.

At least 19 recordsLinked to original sources
arXiv2026

Limit theorems for random Motzkin paths near boundary

We consider Motzkin paths of length $L$, not fixed at zero at both end points, with constant weights on the edges and general weights on the end points. We investigate, as the length $L$ tends to infinity, the limit behaviors of (a) boundary measures induced by the weights on both end points and (b) the segments of the sampled Motzkin path viewed as a process starting from each of the two end points, referred to as boundary processes. Our first result concerns the case when the induced boundary measures have finite first moments. Our second result concerns when the boundary measure on the right end point is a generalized geometric measure with parameter $ρ_1\ge 1$, so that this is an infinite measure and yet it induces a probability measure for random Motzkin path when $ρ_1$ is not too large. The two cases under investigation reveal a phase transition. In particular, we show that the limit left boundary processes in the two cases have the same transition probabilities as random walks conditioned to stay non-negative.

math.PR↗
arXiv2026

Second-order fluctuations for a phase transition in random partitions

In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameters $α\in(0,1)$ and $θ>-α$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\{1,\ldots,n\}$ and consider $j=j_n\to\infty$ as $n\to\infty$. They identified a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\sim rn^{α/(1+α)}$ for some $r>0$. A natural next question is to understand the corresponding second-order fluctuations. We establish second-order limit theorems in the critical regime and, under an additional rate condition in the subcritical regime ($j_n\ll n^{α/(1+α)}/(\log\log n)^{1/(1+α)}$), for the counting process $(C_{j_n}(n(1+t/j_n)_+))_{t\in\mathbb R}$. In the subcritical regime, after appropriate normalization, the limit is a stationary Ornstein--Uhlenbeck Gaussian process, whereas in the critical regime the limit is a stationary $M/M/\infty$ queue. We also establish a more refined point-process convergence in the critical regime. We first establish these results for the more general Karlin infinite urn model and then adapt the analysis to the Chinese restaurant process. For the latter model, most of our limit theorems are established in the quenched sense.

math.PR↗
arXiv2026

On central limit theorems for Ewens-Pitman model

We establish a quenched functional central limit theorem for the total number of components of random partitions induced by a Chinese restaurant process with parameters $(α,θ), α\in(0,1), θ>-α$. With $P_j$ denoting the asymptotic frequency of the $j$-th table, it is well-known that the component count has the same law as the occupancy count of an infinite urn scheme with sampling frequencies being $(P_j)_{j\in\mathbb N}$. Our analysis follows this approach and is based on earlier results of Karlin (1967) and Durieu and Wang (2016). In words, our result reveals that the fluctuations of the component count consist of two parts, one due to the sampling effect given the asymptotic frequencies $(P_j)_{j\in\mathbb N}$, the other due to the fluctuations of the random asymptotic frequencies, and in the limit the fluctuations of the two parts are conditionally independent given the $α$-diversity. Our result strengthens a recent central limit theorem obtained by Bercu and Favaro (2024) via a different method.

math.PR↗
arXiv2025

A family of log-correlated Gaussian processes

A family of log-correlated Gaussian processes indexed by metric spaces is introduced, when the metric is conditionally negative definite. These processes arise as the limit of bi-fractional Brownian motions indexed by $(H,K)$ scaled by $K^{-1/2}$ as $K\downarrow 0$ with $H\in(0,1/2]$ fixed. When the metric is in addition a measure definite kernel, stochastic-integral representations of the generalized processes when evaluated at a test function are provided. The introduced processes are also shown to be the scaling limits of certain aggregated models.

math.PR↗
arXiv2025

Limit fluctuations of stationary measure of totally asymmetric simple exclusion process with open boundaries on the coexistence line

We describe limit fluctuations of the height function for the open TASEP on the coexistence line under the stationary measure. It is known that the height function satisfies a law of large numbers as the number of sites $n$ goes to infinity which at the coexistence line is exotic in the sense that the first-order limit is random. Here, we study the functional central limit theorem: we show that with a random centering and normalized by $\sqrt n$, the second-order limit of the height functions is a (random) mixture of two independent Brownian motions.

math.PR↗
arXiv2025

PhotoArtAgent: Intelligent Photo Retouching with Language Model-Based Artist Agents

Photo retouching is integral to photographic art, extending far beyond simple technical fixes to heighten emotional expression and narrative depth. While artists leverage expertise to create unique visual effects through deliberate adjustments, non-professional users often rely on automated tools that produce visually pleasing results but lack interpretative depth and interactive transparency. In this paper, we introduce PhotoArtAgent, an intelligent system that combines Vision-Language Models (VLMs) with advanced natural language reasoning to emulate the creative process of a professional artist. The agent performs explicit artistic analysis, plans retouching strategies, and outputs precise parameters to Lightroom through an API. It then evaluates the resulting images and iteratively refines them until the desired artistic vision is achieved. Throughout this process, PhotoArtAgent provides transparent, text-based explanations of its creative rationale, fostering meaningful interaction and user control. Experimental results show that PhotoArtAgent not only surpasses existing automated tools in user studies but also achieves results comparable to those of professional human artists.

cs.CV↗
arXiv2025

A functional central limit theorem for weighted occupancy processes of the Karlin model

A functional central limit theorem is established for weighted occupancy processes of the Karlin model. The weighted occupancy processes take the form of, with $D_{n,j}$ denoting the number of urns with $j$-balls after the first $n$ samplings, $\sum_{j=1}^na_jD_{n,j}$ for a prescribed sequence of real numbers $(a_j)_{j\in\mathbb N}$. The main applications are limit theorems for random permutations induced by Chinese restaurant processes with $(α,θ)$-seating with $α\in(0,1), θ>-α$. An example is briefly mentioned here, and full details are provided in an accompanying paper.

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

Limit theorems for random permutations induced by Chinese restaurant processes

We investigate the random permutation matrices induced by the Chinese restaurant processes with $(α,θ)$-seating. When $α=0,θ>0$, the permutations are those following Ewens measures on symmetric groups, and have been extensively studied in the literature. Here, we consider $α\in(0,1)$ and $θ>-α$. In an accompanying paper, a functional central limit theorem is established for partial sum of weighted cycle counts in the form of $\sum_{j=1}^n a_jC_{n,j}$, where $C_{n,j}$ is the number of $j$-cycles of the permutation matrix of size $n$. Two applications are presented. One is on linear statistics of the spectrum, and the other is on the characteristic polynomials outside the unit circle.

math.PR↗
arXiv2024

On the dual representations of Laplace transforms of Markov processes

We provide a general framework for dual representations of Laplace transforms of Markov processes. Such representations state that the Laplace transform of a finite-dimensional distribution of a Markov process can be expressed in terms of a Laplace transform involving another Markov process, but with coefficients in the Laplace transform and time indices of the process interchanged. Dual representations of Laplace transforms have been used recently to study open ASEP and to describe stationary measures of the open KPZ equation. Our framework covers both recently discovered examples in the literature and several new ones, involving general Lévy processes and certain birth-and-death processes.

math.PR↗
arXiv2024

A remarkable example on clustering of extremes for regularly-varying stochastic processes

The stable-regenerative multiple-stable model has been shown recently to have distinct candidate extremal index and extremal index. To understand further this rare phenomenon, two more results are established here for the double-stable model. The first is the convergence of point processes for the clusters of extremes, enhancing the previous result on the weak convergence of random sup-measures. Most interestingly, the second result reveals a new phase transition at the mesoscopic level when computing the asymptotic exceedance probability over a block, $\mathbb P(\max_{k=1,\dots,d_n} X_k>b_n)$, as $n\to\infty$. Here, the mesoscopic level is referred to the fact that the block size $d_n$ is allowed to grow at the rate $n^ρ$ with $ρ\in[0,1]$, while the threshold $b_n$ is such that $\mathbb P(X_1>b_n)\sim 1/n$. The recently discovered discrepancy between the candidate extremal index and the extremal index is shown to be just a reflection of this phase transition that is prohibited by the anticlustering condition.

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↗
arXiv2023

Phase transition for extremes of a family of stationary multiple-stable processes

We investigate a family of multiple-stable processes that may exhibit either long-range or short-range dependence, depending on the parameters. There are two parameters for the processes, the memory parameter $β\in(0,1)$ and the multiplicity parameter $p\in\mathbb N$. We investigate the macroscopic limit of extremes of the process, in terms of convergence of random sup-measures, for the full range of parameters. Our results show that (i) the extremes of the process exhibit long-range dependence when $β_p := pβ-p+1\in(0,1)$, with a new family of random sup-measures arising in the limit, (ii) the extremes are of short-range dependence when $β_p<0$, with independently scattered random sup-measures arising in the limit, and (iii) there is a delicate phase transition at the critical regime $β_p = 0$.

math.PR↗
arXiv2022

Tail processes for stable-regenerative multiple-stable model

We investigate a family of discrete-time stationary processes defined by multiple stable integrals and renewal processes with infinite means. The model may exhibit behaviors of short-range or long-range dependence, respectively, depending on the parameters. The main contribution is to establish a phase transition in terms of the tail processes that characterize local clustering of extremes. Moreover, in the short-range dependence regime, the model provides an example where the extremal index is different from the candidate extremal index.

math.PR↗
arXiv2022

Markov processes related to the stationary measure for the open KPZ equation

We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval $[0,1]$ in terms of a Markov process $Y$, which is a Doob's $h$ transform of the Brownian motion killed at an exponential rate. Our work builds on a recent formula of Corwin and Knizel which expresses the multipoint Laplace transform of the stationary solution of the open KPZ in terms of another Markov process $\mathbb T$: the continuous dual Hahn process with Laplace variables taking on the role of time-points in the process. The core of our approach is to prove that the Laplace transforms of the finite dimensional distributions of $Y$ and $\mathbb T$ are equal when the time parameters of one process become the Laplace variables of the other process and vice versa.

math.PR↗
arXiv2022

From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval

Barraquand and Le~Doussal introduced a family of stationary measures for the (conjectural) KPZ fixed point on an interval with Neumann boundary conditions, and predicted that they arise as scaling limit of stationary measures of all models in the KPZ universality class on an interval. In this paper, we show that the stationary measures for KPZ fixed point on an interval arise as the scaling limits of the height increment processes for the open asymmetric simple exclusion process in the steady state, with parameters changing appropriately as the size of the system tends to infinity.

math.PR↗
arXiv2022

A conditional scaling limit of the KPZ fixed point with height tending to infinity at one location

We consider the asymptotic behavior of the KPZ fixed point $\{\mathsf H(x,t)\}_{x\in\mathbb R, t>0}$ conditioned on $\mathsf H(0,T)=L$ as $L$ goes to infinity. The main result is a conditional limit theorem for the fluctuations of $\mathsf H$ in the region near the line segment connecting the origin $(0,0)$ and $(0,T)$ for both step and flat initial conditions. The limit random field can be represented as a functional of two independent Brownian bridges, and in addition the limit random field depends also on the initial law of the KPZ fixed point. In particular for temporal fluctuations, the limit process indexed by line segment between $(0,0)$ and $(0,T)$, when the KPZ is with step initial condition, has the law of the minimum of two independent Brownian bridges; and when the KPZ is with flat initial condition the limit process has the law of the minimum of two independent Brownian bridges, each in addition perturbed by a common Gaussian random variable. For spatial-temporal fluctuations, the conditional limit theorem sheds light on the asymptotic behaviors of the point-to-point geodesic of the directed landscape conditioned on its length and as the length tends to infinity.

math.PR↗