arXiv subjects

Chien-Hao Huang

Publications and source records attributed to Chien-Hao Huang.

7 recordsLinked to original sources

arXiv2023

Nonsymmetric examples for Gaussian correlation inequalities

In this paper, we compare two variances of maxima of $N$ standard Gaussian random variables. One is a sequence of $N$ i.i.d. standard Gaussians, and the other one is $N$ standard Gaussians with covariances $σ_{1,2}=ρ\in(0,1)$ and $σ_{i,j}=0$, for other $ i\neq j$. It turns out that we need to discuss the covariance of two functions with respect to multivariate Gaussian distributions. Gaussian correlation inequalities hold for many symmetric (with respect to the origin) cases. However, in our case, the max function and its derivatives are not symmetric about the origin. We have two main results in this paper. First, we prove a specific case for a convex/log-concave correlation inequality for the standard multivariate Gaussian distribution. The other result is that the variance of maxima of standard Gaussians with $σ_{1,2}=ρ\in(0,1)$, while $σ_{i,j}=0$, for other $ i\neq j$, is larger than the variance of maxima of independent standard Gaussians. This implies that the variance of maxima of $N$ i.i.d. standard Gaussians is decreasing in $N$.

math.PR↗

arXiv2022

One-dimensional polymers in random environments: stretching vs. folding

In this article we study a \emph{non-directed polymer model} on $\mathbb Z$, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of "rewards" (or penalities) $βω_x -h$ sitting on the range of the random walk, where $(ω_x)_{x\in \mathbb Z}$ are i.i.d.\ random variables (the disorder), and where $β\geq 0$ (disorder strength) and $h\in \mathbb{R}$ (external field) are two parameters. When $β=0,h>0$, this corresponds to a random walk penalized by its range; when $β>0, h=0$, this corresponds to the "standard" polymer model in random environment, except that it is non-directed. In this work, we allow the parameters $β,h$ to vary according to the length of the random walk, and we study in detail the competition between the \emph{stretching effect} of the disorder, the \emph{folding effect} of the external field (if $h\ge 0$), and the \emph{entropy cost} of atypical trajectories. We prove a complete description of the (rich) phase diagram. For instance, in the case $β>0, h=0$ of the non-directed polymer, if $ω_x$ ha a finite second moment, we find a transversal fluctuation exponent $ξ=2/3$, and we identify the limiting distribution of the rescaled log-partition function.

math.PR↗

arXiv2020

Social Distancing 2.0 with Privacy-Preserving Contact Tracing to Avoid a Second Wave of COVID-19

How to avoid a second wave of COVID-19 after reopening the economy is a pressing question. The extremely high basic reproductive number $R_0$ (5.7 to 6.4, shown in new studies) of SARS-CoV-2 further complicates the challenge. Here we assess effects of Social distancing 2.0, i.e. proximity alert (to maintain inter-personal distance) plus privacy-preserving contact tracing. To solve the dual task, we developed an open source mobile app. The app uses a Bluetooth-based, decentralized contact tracing platform over which the anonymous user ID cannot be linked by the government or a third party. Modelling results show that a 50\% adoption rate of Social distancing 2.0, with privacy-preserving contact tracing, would suffice to decrease the $R_0$ to less than 1 and prevent the resurgence of COVID-19 epidemic.

cs.SI↗

arXiv2019

The scaling limits for Wiener sausages in random environments

We consider the statistical mechanics of a random polymer with random walks and disorders in $\mathbb{Z}^d$. The walk collects random disorders along the way and gets nothing if it visits the same site twice. In the continuum and weak disorder regime, the partition function as a random variable converges weakly to a Wiener Chaos expansion when the dimension is lower than the critical dimension, which is four. A finite temperature case in one dimension is also discussed. The last case suggests that the end-point behavior of the polymer is $t^{2/3}$.

math.PR↗

arXiv2015

On the speed of the one-dimensional polymer in the large range regime

We consider a Hamiltonian involving the range of the simple random walk and the Wiener sausage so that the walk tends to stretch itself. This Hamiltonian can be easily extended to the multidimensional cases, since the Wiener sausage is well-defined in any dimension. In dimension one, we give a formula for the speed and the spread of the endpoint of the polymer path. It can be easily showed that if the self-repelling strength is stronger, the end point is going away faster. This strict monotonicity of speed has not been proven in the literature for the one-dimensional case.

math.PR↗

arXiv2014

Random potentials for pinning models with Laplacian interactions

We consider a statistical mechanics model for biopolymers. Sophisticated polymer chains, such as DNA, have stiffness when they stretch chains. The Laplacian interaction is used to describe the stiffness. Also, the surface between two media has an attraction force, and the force will pull the chain back to the surface. In this paper, we deal with the random potentials when the monomers interact with the random media. Although these models are different from the pinning models studied before, the result about the gap between the annealed critical point and the quenched critical point stays the same.

math.PR↗

arXiv2012

Random potentials for pinning models with \nabla and Δinteractions

We consider two models for biopolymers, the $\nabla$ interaction and the $Δ$ one, both with the Gaussian potential in the random environment. A random field $φ:{0,1,...,N}\rightarrow \Bbb{R}^d$ represents the position of the polymer path. The law of the field is given by $\exp(-\sum_i\frac{|\nablaφ_i|^2}{2})$ where $\nabla$ is the discrete gradient, and by $\exp(-\sum_i\frac{|Δφ_i|^2}{2})$ where $Δ$ is the discrete Laplacian. For every Gaussian potential $\frac{|\cdot|^2}{2}$, a random charge is added as a factor: $(1+βω_i)\frac{|\cdot|^2}{2}$ with $\Bbb{P}(ω_i=\pm 1)=1/2$ or $\exp(βω_i)\frac{|\cdot|^2}{2}$ with $ω_i$ obeys a normal distribution. The interaction with the origin in the random field space is considered. Each time the field touches the origin, a reward $ε\geq 0$ is given. Although these models are quite different from the pinning models studied in Giacomin (2007), the result about the gap between the annealed critical point and the quenched critical point stays the same.

math.PR↗