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All entries for Friday 22 June 2012

## June 22, 2012

### URSS Action Plan

### Summary

Summary of 'elevator pitches'

George, who is from Medical school, investigates the causality between sleep and weight.

Hong, who is from Engineering department, tries to find cheaper method to produce metal Ti.

Lewis, who is from Physics department, investigates if the two transmittors of a mobile phone can be combined into one.

Adrian, who is from Mathematics department, studies the device which can help the deaf to hear by stimulating the certain nervous cells in the brain.

My project is a study of the cover times of random walks on finite graphs. The specific aim will be to understand random walks better with the help of theory on electric network and extend existing explicit results for the simple random walk (that is, when the walker is equally likely to jump to any neighbour at a given time step) to more general models, such as when edges are considered to be weighted, or when transition probabilities are selected randomly. Then with the help of the explicit formula, some of the properties which are related to the cover times in the general models can be analysed asymptotically, e.g. the probability that the last vertex visited is the vertex i. Furthermore, I will try to see if the results obtained are useful for more complicated graphs.

The activities of my project involve reading textbooks and papers, analysis of chains, derivation of formula and simulation.

The reason is that the project is highly related to what I am interested in, i.e. the Markov process, random walk and analysing some graph properties with the help of probability.

### Actions

### Researcher Development Goal 1

#### What is the challenge or skill that I would like to develop?

Understanding the connections between random walks and electric networks.

#### What am I going to do to achieve this?

To read textbooks and papers, try to solve specific questions on this topic and ask my supervisor questions when I cannot understand after my own thinking.

#### Why have I chosen this particular objective?

Because the electric networks give a good intuition for analysing random walks. It is important to understand the connections and use electric networks as a tool to analyse random walks.

#### Is this achievable within the timeframe and resources available?

I have some background knowledges about electric networks from my A-level, so it will not cost me much time to refresh my memory. And my supervisor listed a set of related textbooks and papers, which can be found in the library or online.

#### When will I achieve this by?

I think this will be achieved in the early stage of the project( probably end of the second week), since further analysis requires this as background.

#### How am I going to know that I have been successful?

If I can clearly explain the main ideas to my friends who do not know the topic and successfully apply them in my further analysis of random walks, then it is enough to know that I understand the their connections well.

### Researcher Development Goal 2

#### What is the challenge or skill that I would like to develop?

To derive explicit formula for cover times of random conductance model on a polygon.

#### What am I going to do to achieve this?

To achieve this, I shall first understand the cover times of simple random walk on a polygon and then with the help of electric networks and Markov property to derive the explicit formula.

#### Why have I chosen this particular objective?

This is one of the main aims of my project and it is important for further analysis of my project.

#### Is this achievable within the timeframe and resources available?

My supervisor provided me a sufficient set of related textbooks and papers and the formula is already derived for simple random work on a polygon.

#### When will I achieve this by?

I think this will be achieved in the middle stage of the project (probably by the end of the third or fourth week).

#### How am I going to know that I have been successful?

If the formula derived is true at least for the simple random walk and the arguments through the deduction are logical and plausible, then I am very likely to find the correct formula. Also a simulation can check the correctness of my formula.