Randomly oriented polymers on R^2 in a spatial white noise environment

Professor Introduction

R. J | Ph.D. in Mathematics

Home Institute：University of Oxford

[ Research Interests ] Stochastic analysis, financial mathematics, stochastic partial differential equations, stochastic quantization
[ Awards ] Silver medal in the Analysis category of the S.-T. Yau College Student Mathematics Contest, multiple team and individual awards

Project Description

This Project provides a rigorous mathematical framework for studying directed polymers in random environments, particularly focusing on the impact of spatial white noise on these polymers within R^2. Directed polymers are a class of models in statistical mechanics, which describe the behavior of polymer chains influenced by random media. In this Project, we will delve into the measure of a directed polymer perturbed by spatial white noise, modeled as a Brownian motion in the continuous function space. Starting from discrete models, we will build up to the study of random walks perturbed by multi-dimensional Gaussian random variables and the corresponding discrete stochastic partial differential equations (SPDEs). The convergence from discrete SPDEs to continuous SPDEs will be analyzed to characterize the random measures of interest. If time permits, we will further explore the general properties of these measures. One of the primary challenges in this project is understanding the impact of the weights influenced by spatial white noise, which exhibits logarithmic growth, on the overall analysis.

Project Keywords

Project Outline

Part 1 : Introduction to Directed Polymers and Random Environments
• Fundamental concepts of directed polymers
• Overview of random environments and spatial white noise
• Basic properties and challenges in analyzing directed polymers in random settings

Part 2 : Discrete Models and Gaussian Perturbations
• Construction of discrete models for directed polymers
• Analysis of random walks perturbed by multi-dimensional Gaussian variables
• Introduction to discrete stochastic partial differential equations (SPDEs)

Part 3 :  From Discrete to Continuous SPDEs
• Convergence of discrete SPDEs to continuous SPDEs
• Mathematical techniques for analyzing convergence
• Characterization of random measures in continuous settings

Part 4 : Properties of Random Measures
• General properties of random measures influenced by spatial white noise
• Impact of weight functions with logarithmic growth
• Advanced topics in stochastic analysis related to directed polymers

Part 5 : Research and Practical Applications
• Current research trends in directed polymers and random environments
• Open problems and future directions in the field
• Student-led research projects and presentations

Suitable for

• High School Students preparing for research in stochastic partial differential equations andmathematics areas
• Advanced undergraduate and graduate students in mathematics, particularly those specializing in stochastic processes and mathematical physics