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Xiaofeng Xue

Publications and source records attributed to Xiaofeng Xue.

At least 37 records · Page 2Linked to original sources
arXiv2022

Moderate deviations of hitting times of a family of density-dependent Markov chains

In this paper we are concerned with hitting times of a family of density-dependent Markov chains. A moderate deviation principle of the hitting time is given. The proof of the main theorem relies heavily on moderate deviations of density-dependent Markov chains given in \cite{Xue2021} and upper bounds of large deviations of Markov processes given in \cite{Dupuis1991}. An analogue moderate deviation of the hitting time of the diffusion approximation of the density-dependent Markov chain introduced in \cite{Ethier1986} is also given.

math.PR↗
arXiv2022

Hydrodynamics of $N$-urn susceptible-infected-removed epidemics

In this paper we are concerned with $N$-urn susceptible-infected-removed epidemics, where each urn is in one of three states, namely `susceptible', `infected' and `removed'. We assume that recovery rates of infected urns and infection rates between infected and susceptible urns are all coordinate-dependent. We show that the hydrodynamic limit of our model is driven by a deterministic $(C[0, 1])^\prime$-valued process with density which is the solution to a nonlinear $C[0, 1]$-valued ordinary differential equation consistent with a mean-field analysis. We further show that the fluctuation of our process is driven by a generalized Ornstein-Uhlenbeck process. A key step in proofs of above main results is to show that sates of different urns are approximately independent as $N\rightarrow+\infty$.

math.PR↗
arXiv2022

Moderate Deviations for the current and Tagged Particle in Symmetric Simple Exclusion Processes

We prove moderate deviation principles for the tagged particle position and current in one-dimensional symmetric simple exclusion processes. There is at most one particle per site. A particle jumps to one of its two neighbors at rate $1/2$, and the jump is suppressed if there is already one at the target site. We distinguish one particular particle which is called the tagged particle. We first establish a variational formula for the moderate deviation rate functions of the tagged particle positions based on moderate deviation principles from hydrodynamic limit proved by Gao and Quastel \cite{gao2003moderate}. Then we construct a minimizer of the variational formula and obtain explicit expressions for the moderate deviation rate functions.

math.PR↗
arXiv2022

Hydrodynamics of a class of $N$-urn linear systems

In this paper we are concerned with hydrodynamics of a class of $N$-urn linear systems, which include voter models, pair-symmetric exclusion processes and binary contact path processes on $N$ urns as special cases. We show that the hydrodynamic limit of our process is driven by a $\left(C[0,1]\right)^\prime$-valued linear ordinary differential equation and the fluctuation of our process, i.e, central limit theorem from the hydrodynamic limit, is driven by a $\left(C[0, 1]\right)^\prime$-valued Ornstein-Uhlenbeck process. To derive above main results, we need several replacement lemmas. An extension in linear systems of Chapman-Kolmogorov equation plays key role in proofs of these replacement lemmas.

math.PR↗
arXiv2021

Moderate deviations of generalized $N$-urn Ehrenfest models

This paper is a further investigation of the generalized $N$-urn Ehrenfest model introduced in \cite{Xue2020}. A moderate deviation principle from the hydrodynamic limit of the model is derived. The proof of this main result follows a routine procedure introduced in \cite{Kipnis1989}, where a replacement lemma plays the key role. To prove the replacement lemma, the large deviation principle of the model given in \cite{Xue2020} is utilized.

math.PR↗
arXiv2020

Hydrodynamics of the generalized $N$-urn Ehrenfest model

In this paper we are concerned with a generalized $N$-urn Ehrenfest model, where balls keeps independent random walks between $N$ boxes uniformly laid on $[0, 1]$. After a proper scaling of the transition rates function of the aforesaid random walk, we derive the hydrodynamic limit of the model, i.e., the law of large numbers which the empirical measure of the model follows, under an assumption where the initial number of balls in each box independently follows a Poisson distribution. We show that the empirical measure of the model converges weakly to a deterministic measure with density driven by an integral equation. Furthermore, we derive non-equilibrium fluctuation of the model, i.e, the central limit theorem from the above hydrodynamic limit. We show that the non-equilibrium fluctuation of the model is driven by a measure-valued time-inhomogeneous generalized O-U process. At last, we prove a large deviation principle from the hydrodynamic limit under an assumption where the transition rates function from $[0, 1]\times [0, 1]$ to $[0, +\infty)$ of the aforesaid random walk is a product of two marginal functions from $[0, 1]$ to $[0, +\infty)$.

math.PR↗
arXiv2020

Non-equilibrium Fluctuations of the Weakly Asymmetric Normalized Binary Contact Path Process

This paper is a further investigation of the problem studied in \cite{xue2020hydrodynamics}, where the authors proved a law of large numbers for the empirical measure of the weakly asymmetric normalized binary contact path process on $\mathbb{Z}^d,\, d \geq 3$, and then conjectured that a central limit theorem should hold under a non-equilibrium initial condition. We prove that the aforesaid conjecture is true when the dimension $d$ of the underlying lattice and the infection rate $λ$ of the process are sufficiently large.

math.PR↗
arXiv2020

Statistical inference for unknown parameters of stochastic SIS epidemics on complete graphs

In this paper, we are concerned with the stochastic susceptible-infectious-susceptible (SIS) epidemic model on the complete graph with $n$ vertices. This model has two parameters, which are the infection rate and the recovery rate. By utilizing the theory of density-dependent Markov chains, we give consistent estimations of the above two parameters as $n$ grows to infinity according to the sample path of the model in a finite time interval. Furthermore, we establish the central limit theorem (CLT) and the moderate deviation principle (MDP) of our estimations. As an application of our CLT, reject regions of hypothesis testings of two parameters are given. As an application of our MDP, confidence intervals with lengths converging to $0$ while confidence levels converging to $1$ are given as $n$ grows to infinity.

math.PR↗
arXiv2020

Moderate Deviations for the SSEP with a Slow Bond

We consider the one dimensional symmetric simple exclusion process with a slow bond. In this model, particles cross each bond at rate $N^2$, except one particular bond, the slow bond, where the rate is $N$. Above, $N$ is the scaling parameter. This model has been considered in the context of hydrodynamic limits, fluctuations and large deviations. We investigate moderate deviations from hydrodynamics and obtain a moderate deviation principle.

math.PR↗
arXiv2020

Moderate deviations of density-dependent Markov chains

The density-dependent Markov chain (DDMC) introduced in \cite{Kurtz1978} is a continuous time Markov process applied in fields such as epidemics, chemical reactions and so on. In this paper, we give moderate deviation principles of paths of DDMC under some generally satisfied assumptions. The proofs for the lower and upper bounds of our main result utilize an exponential martingale and a generalized version of Girsanov's theorem. The exponential martingale is defined according to the generator of DDMC.

math.PR↗
arXiv2019

High-speed Privacy Amplification Scheme using GMP in Quantum Key Distribution

Privacy amplification (PA) is the art of distilling a highly secret key from a partially secure string by public discussion. It is a vital procedure in quantum key distribution (QKD) to produce a theoretically unconditional secure key. The throughput of PA has become a bottleneck of the high-speed discrete variable QKD (DV-QKD) system. In this paper, a high-speed modular arithmetic hash PA scheme with GNU multiple precision (GMP) arithmetic library is presented. This scheme is implemented on two different central processing unit (CPU) platforms. The experimental results demon-strate that the throughput of this scheme achieves 260Mbps on the block size of 10^6 and 140Mbps on the block size of 10^8. This is the highest-speed recorded PA scheme on CPU platform to the author's knowledge.

quant-ph↗
arXiv2019

Hydrodynamics of the Binary Contact Path Process

In this paper we are concerned with the binary contact path process introduced in \cite{Gri1983} on the lattice $\mathbb{Z}^d$ with $d\geq 3$. Our main result gives a hydrodynamic limit of the process, which is the solution to a heat equation. The proof of our result follows the strategy introduced in \cite{kipnis+landim99} to give hydrodynamic limit of the SEP model with some details modified since the states of all vertices are not uniformly bounded for the binary contact path process. In the modifications, the theory of the linear system introduced in \cite{Lig1985} is utilized.

math.PR↗
arXiv2018

Two limit theorems for the high-dimensional two-stage contact process

In this paper we are concerned with the two-stage contact process introduced in \cite{Krone1999} on a high-dimensional lattice. By comparing this process with an auxiliary model which is a linear system, we obtain two limit theorems for this process as the dimension of the lattice grows to infinity. The first theorem is about the upper invariant measure of the process. The second theorem is about asymptotic behavior of the critical value of the process. These two theorems can be considered as extensions of their counterparts for the basic contact processes proved in \cite{Grif1983} and \cite{Schonmann1986}.

math.PR↗
arXiv2017

The critical infection rate of the high-dimensional two-stage contact process

In this paper we are concerned with the two-stage contact process on the lattice $\mathbb{Z}^d$ introduced in \cite{Krone1999}. We gives a limit theorem of the critical infection rate of the process as the dimension $d$ of the lattice grows to infinity. A linear system and a two-stage SIR model are two main tools for the proof of our main result.

math.PR↗
arXiv2017

The survival probability of the high-dimensional contact process with random vertex weights on the oriented lattice

This paper is a further study of Reference \cite{Xue2015}. We are concerned with the contact process with random vertex weights on the oriented lattice. Our main result gives the asymptotic behavior of the survival probability of the process conditioned on only one vertex is infected at $t=0$ as the dimension grows to infinity. A SIR model and a branching process with random vertex weights are the main auxiliary tools for the proof of the main result.

math.PR↗
arXiv2017

Survival probabilities of high-dimensional stochastic SIS and SIR models with random edge weights

In this paper, we are concerned with the stochastic SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered) models on high-dimensional lattices with random edge weights, where a susceptible vertex is infected by an infectious neighbor at rate proportional to the weight on the edge connecting them. All the edge weights are assumed to be i.i.d.. Our main result gives mean field limits for survival probabilities of the two models as the dimension grows to infinity, which extends the main conclusion given in \cite{Xue2017} for classic stochastic SIS model.

math.PR↗