arXiv subjects

Guanglin Rang

Publications and source records attributed to Guanglin Rang.

6 recordsLinked to original sources

arXiv2026

Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations

We study a Brownian directed polymer in a centered Gaussian environment that is white in time and colored in space having long-range spatial correlations. For product-type covariances $Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-\alpha_j}$, with $\alpha_j\in(0,1)$ and $\kappa=\sum_j\alpha_j$, we identify the disorder transition at the marginal value $\kappa=2$. For $\kappa>2$, weak disorder holds at sufficiently small inverse temperature; for $\kappa<2$, the quenched free energy $p(\beta)$ satisfies $-p(\beta)\asymp\beta^{4/(2-\kappa)}$ as $\beta\downarrow0$. For $\kappa=2$, strong disorder holds for every $\beta>0$, while $p(\beta)=0$ for all sufficiently small $\beta$, so $\beta_c=0<\bar\beta_c$. We also consider the radial covariance cases, where $ Q(x)\asymp(1+|x|)^{-\vartheta}$, when $d\ge3,\vartheta=2$ and $d=2,\vartheta\ge2$, which was left unanswered in Lacoin~\cite{Lacoin2011}. When $d=2$, we get $\ln(-p(\beta))\asymp-\beta^{-2}$ for $\vartheta>2$ and $\ln(-p(\beta))\asymp-\beta^{-1}$ for $\vartheta=2$. The proofs consist of replica coupling, Feynman--Kac variational formula, overlap methods, and continuous-space fractional moments with ordered Wiener-chaos changes of measure.

math.PR↗

arXiv2026

Scaling limit of 1+1 dimensional directed polymer with power-law tail and spatial correlated noise

We study a $(1+1)$-dimensional directed polymer in a spatially correlated random environment generated by power-law tail variables: $\omega(i,x)=\sum_{y\in\mathbb Z}\psi_{y-x}\xi(i,y), \psi_y\sim \lambda_r |y|^{-r}, r\in(1/2,1)$, where the variables $\xi(i,y)$ are i.i.d. and have a regularly varying right tail with exponent $\alpha>2$. The spatial covariance of the environment has long-range decay with Hurst parameter $H=\frac32-r\in(1/2,1)$. We identify the limiting fluctuations of the log-partition function in the intermediate disorder regime and show that the critical tail exponent is $\alpha_c=\frac{3}{H}=\frac{6}{3-2r}$. When $\alpha>\alpha_c$, the model has the same scaling limits as the corresponding Gaussian spatially correlated polymer: if $\beta_NN^{H/2}\to\beta\in(0,\infty)$, the centered log-partition function converges to the logarithm of the solution of the stochastic heat equation driven by fractional spatial noise; if $\beta_NN^{H/2}\to0$, its normalized fluctuation converges to a centered Gaussian law. In the regime $2<\alpha\le\alpha_c$, at the scale $\beta_NN^{H/2}=\beta N^{H/2}/l(N^{3/2})$, the log-partition function still satisfies Gaussian fluctuation. The main ingredient is a truncation comparison argument adapted to long-range moving-average environments, together with an invariance principle for polynomial chaos. Due to the non-locality of the environments, we perform a far-near field analysis, as well as multiscale analysis, to prove that the truncated version does not change the log-partition function at the corresponding scales.

math.PR↗

arXiv2026

Normal approximation for the polynomial functionals of correlated random field sampling along random walk path in dimension $1+1$

Let $\xi$ be the stationary occupation field generated by a Poisson system of independent simple symmetric random walks on $\mathbb Z$ in space--time dimension $1+1$. For a finite set $A\subset\mathbb Z$, we consider the classical fixed-region observables $W_N(A)$, the cumulative occupation of $A$ up to time $N$, and $D_N(A)$, the number of distinct particles visiting $A$ up to time $N$. We prove quantitative central limit theorems for both observables, with Wasserstein rate of order $N^{-1/4}$. In addition, we introduce an independent nearest-neighbour random walk $S=(S_n,\,n\ge 0)$ on $\mathbb Z$ with non-zero drift and sample the field along this ballistic path. For a fixed polynomial observable $\varphi(x)=\sum_{j=0}^k \beta_j x^j, \beta_k\neq 0$, of degree $k\in \mathbb N$, we consider the partial sums $Y_{N,\varphi}=\sum_{n=1}^N \varphi(\xi(n,S_n)).$ We prove a Wasserstein bound of order $N^{-1/2}$ for the normal approximation of the standardized $Y_{N,\varphi}$. To the best of our knowledge, this is the first quantitative normal approximation result for polynomial functionals of the Poisson occupation field sampled along a random walk path. The drift induces an effective decorrelation of the sampled environment, leading to a substantial improvement over fixed-region sampling. The proofs rely on a representation of $\xi$ as a Poisson functional on path space and on the Malliavin--Stein method for Poisson functionals.

math.PR↗

arXiv2026

Local linearization for the nonlinear damped stochastic Klein-Gordon equation

For the $1+1$ dimensional nonlinear damped stochastic Klein-Gordon equation driven by space-time white noise, we prove that the second-order increments of the solution can be approximated, after scaling with the diffusion coefficient, by those of the corresponding linearized stochastic Klein-Gordon equation. This extends the result of Huang et al. \cite{HOO2024} for the stochastic wave equation. A key difficulty arises from the more complex structure of the Green function, which we overcome by means of subtle analytical estimates. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

math.PR↗

arXiv2022

Scaling limit of a long-range random walk in time-correlated random environment

This paper concerns a long-range random walk in random environment in dimension $1+1$, where the environmental disorder is independent in space but has long-range correlations in time. We prove that two types of rescaled partition functions converge weakly to the Stratonovich solution and the It\^o-Skorohod solution respectively of a fractional stochastic heat equation with multiplicative Gaussian noise which is white in space and colored in time.

math.PR↗

arXiv2017

From Directed Polymers in Spatial-correlated Environment to Stochastic Heat Equations Driven by Fractional Noise in 1 + 1 Dimensions

We consider the limit behavior of partition function of directed polymers in random environment represented by linear model instead of a family of i.i.d.variables in $1+1$ dimensions. Under the assumption that the correlation decays algebraically, using the method developed in [Ann. Probab., 42(3):1212-1256, 2014], under a new scaling we show the scaled partition function as a process defined on $[0,1]\times\RR$, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The Hurst parameter is determined by the correlation exponent of the random environment. Here multiple It\^{o} integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.

math.PR↗