arXiv subjects

Yuchen Liao

Publications and source records attributed to Yuchen Liao.

14 recordsLinked to original sources

arXiv2026

Port-Hamiltonian Systems with Dissipation Potential: Modelling and Trajectory Tracking Control

Port-Hamiltonian systems (PHS) and interconnection and damping assignment passivity-based control (IDA-PBC) have achieved broad success in modelling and stabilisation of physical systems. However, the absence of a dedicated scalar potential for the momentum channel forces any modification of the momentum-dependent dynamics to proceed indirectly through the interconnection and damping matrices, rendering the matching partial differential equation (PDE) difficult to solve and complicating extensions to trajectory tracking. This paper proposes a port-Hamiltonian system with dissipation potential (PHS-DP), in which the damping matrix is replaced by scalar convex dissipation potentials, providing independent scalar objects for the momentum and auxiliary state channels and restoring the variational symmetry between stored and dissipated energy. Building on this framework, Dual Potential Shaping Control (DPSC) achieves trajectory tracking by sequentially shaping the potential energy and dissipation potentials without modifying the interconnection structure. Contraction of the closed-loop cascade is established via a hierarchical contraction argument, and the matching condition is satisfied automatically for any admissible choice of shaped potentials, requiring no PDE to be solved. In contrast to existing PDE-free energy shaping approaches, which achieve this by abandoning the port-Hamiltonian closed-loop structure and sacrificing physical interpretability, the proposed framework preserves the interconnection structure and retains a transparent energy-based interpretation at every stage of the design. Validation on a magnetic levitation system demonstrates tracking performance comparable to timed IDA-PBC with substantially reduced design complexity.

eess.SY↗

arXiv2026

A Remark on Downlink Massive Random Access

In downlink massive random access (DMRA), a base station transmits messages to a typically small subset of active users, selected randomly from a massive number of total users. Explicitly encoding the identities of active users would incur a significant overhead scaling logarithmically with the number of total users. Recently, via a random coding argument, Song, Attiah and Yu have shown that the overhead can be reduced to within some upper bound irrespective of the number of total users. In this remark, recognizing that the code design for DMRA is an instance of covering arrays in combinatorics, we show that there exists deterministic construction of variable-length codes that incur an overhead no greater than $1 + log_2 e$ bits.

cs.IT↗

arXiv2026

Periodic KPZ fixed point with general initial conditions

We consider the relaxation-time-scale limit of the periodic totally asymmetric simple exclusion process (PTASEP) with general initial conditions. For every sequence of initial conditions approximating a periodic upper semicontinuous function, we compute the limiting space-time multipoint distributions of the rescaled particle locations and height functions. The resulting finite-dimensional distributions are explicit and form a consistent family, thereby defining a spatially periodic space-time random field. We call this field the periodic KPZ fixed point with the corresponding initial condition. This extends earlier results for PTASEP with special initial conditions and defines the periodic analogue of the KPZ fixed point on the line. The main technical novelty is a pair of new probabilistic representations for the energy function and the characteristic function, the two functions through which the initial condition enters the finite-time PTASEP multipoint distribution formula. Both representations are expressed in terms of a geometric random walk and two stopping times, namely the first hitting time of the initial profile and the first such hitting time at or after one full period, with the latter capturing the periodic geometry.

math.PR↗

arXiv2025

Multipoint distributions of the KPZ fixed point with compactly supported initial conditions

The KPZ fixed point is a universal limiting space-time random field for the Kardar-Parisi-Zhang universality class. While the joint law of the KPZ fixed point at a fixed time has been studied extensively, the multipoint distributions of the KPZ fixed point in the general space-time plane are much less well understood. More explicitly, formulas were only available for the narrow wedge initial condition arXiv:1906.01053, arXiv:1907.09876 and the flat initial condition arXiv:1907.09876 for the multipoint distributions, and the half-Brownian and Brownian initial conditions arXiv:2010.07357v1, arXiv:2504.19975 for the two-point distributions. In this paper, we obtain the first formula for the space-time joint distributions of the KPZ fixed point with general initial conditions of compact support. The formula is obtained through taking $1:2:3$ KPZ scaling limit of the multipoint distribution formulas for the totally asymmetric simple exclusion process (TASEP). A key ingredient is a probabilistic representation, inspired by arXiv:1701.00018, of the kernel encoding the initial condition for TASEP, which was first defined through an implicit characterization in arXiv:1907.09876. Moreover, we also verify that the equal time version of our formula matches the path integral formula in arXiv:1701.00018 for the KPZ fixed point when the initial condition is of compact support.

math.PR↗

arXiv2025

Large deviations for the $q$-deformed polynuclear growth

In this paper, we study large time large deviations for the height function $\mathfrak{h}(x,t)$ of the $q$-deformed polynuclear growth introduced in ABW22 [arXiv:2108.06018]. We show that the upper-tail deviations have speed $t$ and derive an explicit formula for the rate function $Φ_+(μ)$. On the other hand, we show that the lower-tail deviations have speed $t^2$ and express the corresponding rate function $Φ_-(μ)$ in terms of a variational problem. Our analysis relies on distributional identities between the height function $\mathfrak{h}$ and two important measures on the set of integer partitions: the Poissonized Plancherel measure and the cylindric Plancherel measure. Following a scheme developed in DT21 [arXiv:1910.09271], we analyze a Fredholm determinant representation for the $q$-Laplace transform of $\mathfrak{h}(x,t)$, from which we extract exact Lyapunov exponents and through inversion the upper-tail rate function $Φ_+$. The proof of the lower-tail large deviation principle is more subtle and requires several novel ideas which combine classical asymptotic results for the Plancherel measure and log-concavity properties of Schur polynomials. Techniques we develop to characterize the lower-tail are rather flexible and have the potential to generalize to other solvable growth models.

math.PR↗

arXiv2025

Lower tail large deviations of the stochastic six vertex model

In this paper, we study lower tail probabilities of the height function $\mathfrak{h}(M,N)$ of the stochastic six-vertex model. We introduce a novel combinatorial approach to demonstrate that the tail probabilities $\mathbb{P}(\mathfrak{h}(M,N) \ge r)$ are log-concave in a certain weak sense. We prove further that for each $α>0$ the lower tail of $-\mathfrak{h}(\lfloor αN \rfloor, N)$ satisfies a Large Deviation Principle (LDP) with speed $N^2$ and a rate function $Φ_α^{(-)}$, which is given by the infimal deconvolution between a certain energy integral and a parabola. Our analysis begins with a distributional identity from BO17 [arXiv:1608.01564], which relates the lower tail of the height function, after a random shift, with a multiplicative functional of the Schur measure. Tools from potential theory allow us to extract the LDP for the shifted height function. We then use our weak log-concavity result, along with a deconvolution scheme from our earlier paper [arXiv:2307.01179], to convert the LDP for the shifted height function to the LDP for the stochastic six-vertex model height function.

math.PR↗

arXiv2024

Decoupling Contact for Fine-Grained Motion Style Transfer

Motion style transfer changes the style of a motion while retaining its content and is useful in computer animations and games. Contact is an essential component of motion style transfer that should be controlled explicitly in order to express the style vividly while enhancing motion naturalness and quality. However, it is unknown how to decouple and control contact to achieve fine-grained control in motion style transfer. In this paper, we present a novel style transfer method for fine-grained control over contacts while achieving both motion naturalness and spatial-temporal variations of style. Based on our empirical evidence, we propose controlling contact indirectly through the hip velocity, which can be further decomposed into the trajectory and contact timing, respectively. To this end, we propose a new model that explicitly models the correlations between motions and trajectory/contact timing/style, allowing us to decouple and control each separately. Our approach is built around a motion manifold, where hip controls can be easily integrated into a Transformer-based decoder. It is versatile in that it can generate motions directly as well as be used as post-processing for existing methods to improve quality and contact controllability. In addition, we propose a new metric that measures a correlation pattern of motions based on our empirical evidence, aligning well with human perception in terms of motion naturalness. Based on extensive evaluation, our method outperforms existing methods in terms of style expressivity and motion quality.

cs.CV↗

arXiv2023

RSMT: Real-time Stylized Motion Transition for Characters

Styled online in-between motion generation has important application scenarios in computer animation and games. Its core challenge lies in the need to satisfy four critical requirements simultaneously: generation speed, motion quality, style diversity, and synthesis controllability. While the first two challenges demand a delicate balance between simple fast models and learning capacity for generation quality, the latter two are rarely investigated together in existing methods, which largely focus on either control without style or uncontrolled stylized motions. To this end, we propose a Real-time Stylized Motion Transition method (RSMT) to achieve all aforementioned goals. Our method consists of two critical, independent components: a general motion manifold model and a style motion sampler. The former acts as a high-quality motion source and the latter synthesizes styled motions on the fly under control signals. Since both components can be trained separately on different datasets, our method provides great flexibility, requires less data, and generalizes well when no/few samples are available for unseen styles. Through exhaustive evaluation, our method proves to be fast, high-quality, versatile, and controllable. The code and data are available at {https://github.com/yuyujunjun/RSMT-Realtime-Stylized-Motion-Transition.}

cs.CV↗

arXiv2023

Moment Intermittency in the PAM with Asymptotically Singular Noise

Let $ξ$ be a singular Gaussian noise on $\mathbb R^d$ that is either white, fractional, or with the Riesz covariance kernel; in particular, there exists a scaling parameter $ω>0$ such that $c^{ω/2}ξ(c\cdot)$ is equal in distribution to $ξ$ for all $c>0$. Let $(ξ_\varepsilon)_{\varepsilon>0}$ be a sequence of smooth mollifications such that $ξ_\varepsilon\toξ$ as $\varepsilon\to0$. We study the asymptotics of the moments of the parabolic Anderson model (PAM) with noise $ξ_\varepsilon$ as $\varepsilon\to0$, both for large (i.e., $t\to\infty$) and fixed times $t$. This approach makes it possible to study the moments of the PAM with regular and singular noises in a unified fashion, as well as interpolate between the two settings. As corollaries of our main results, we obtain the following: $\textbf{(1)}$ When $ξ$ is subcritical (i.e., $0<ω<2$), our results extend the known large-time moment and tail asymptotics for the Stratonovich PAM with noise $ξ$. Our method of proof clarifies the role of the maximizers of the variational problems (known as Hartree ground states) that appear in these moment asymptotics in describing the geometry of intermittency. We take this opportunity to prove the existence and study the properties of the Hartree ground state with a fractional kernel, which we believe is of independent interest. $\textbf{(2)}$ When $ξ$ is critical or supercritical (i.e., $ω=2$ or $ω>2$), our results provide a new interpretation of the moment blowup phenomenon observed in the Stratonovich PAM with noise $ξ$. That is, we uncover that the latter is related to an intermittency effect that occurs in the PAM with noise $ξ_\varepsilon$ as $\varepsilon\to0$ for $\textit{fixed finite times}$ $t>0$.

math.PR↗

arXiv2023

Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point

We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson-Schensted-Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalising the formulation of Matetski, Quastel and Remenik (Acta Math., 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case.

math.PR↗

arXiv2023

On Spatial Conditioning of the Spectrum of Discrete Random Schrödinger Operators

Consider a random Schrödinger-type operator of the form $H:=-H_X+V+ξ$ acting on a general graph $\mathscr G=(\mathscr V,\mathscr E)$, where $H_X$ is the generator of a Markov process $X$ on $\mathscr G$, $V$ is a deterministic potential with sufficient growth (so that $H$ has a purely discrete spectrum), and $ξ$ is a random noise with at-most-exponential tails. We prove that $H$'s eigenvalue point process is number rigid in the sense of Ghosh and Peres (Duke Math. J. 166 (2017), no. 10, 1789--1858); that is, the number of eigenvalues in any bounded domain $B\subset\mathbb C$ is determined by the configuration of eigenvalues outside of $B$. Our general setting allows to treat cases where $X$ could be non-symmetric (hence $H$ is non-self-adjoint) and $ξ$ has long-range dependence. Our strategy of proof consists of controlling the variance of the trace of the semigroup $\mathrm e^{-t H}$ using the Feynman-Kac formula.

math-ph↗

arXiv2022

Rigidity of the Stochastic Airy Operator

We prove that the spectrum of the stochastic Airy operator is rigid in the sense of Ghosh and Peres (Duke Math. J., 166(10):1789--1858, 2017) for Dirichlet and Robin boundary conditions. This proves the rigidity of the Airy-$β$ point process and the soft-edge limit of rank-$1$ perturbations of Gaussian $β$-Ensembles for any $β>0$, and solves an open problem mentioned in a previous work of Bufetov, Nikitin, and Qiu (Mosc. Math. J., 19(2):217--274, 2019). Our proof uses a combination of the semigroup theory of the stochastic Airy operator and the techniques for studying insertion and deletion tolerance of point processes developed by Holroyd and Soo (Electron. J. Probab., 18:no. 74, 24, 2013).

math.PR↗

arXiv2022

Multi-point distribution of discrete time periodic TASEP

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale $t=O(L^{3/2})$ when the height fluctuations are critically affected by the finite geometry. The assumptions are verified for the step and flat initial conditions. As a corollary we obtain the multi-point distribution of discrete time TASEP on the whole integer lattice $\mathbb{Z}$ by taking the period $L$ large enough so that the finite-time distribution is not affected by the boundary. The large time limit for multi-time distribution for discrete time TASEP on $\mathbb{Z}$ is then obtained for the step initial condition.

math.PR↗

arXiv2020

Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas

We develop a technique for proving number rigidity (in the sense of Ghosh-Peres) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman-Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman-Kac formulas for RSOs with multiplicative white noise by Gorin, Shkolnikov and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $-\frac12Δ+V+ξ$, where $V$ is a deterministic potential and $ξ$ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential $V$, and the singularity of the noise $ξ$.

math-ph↗