*Development Goal*

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## September 14, 2012

### Researcher Development Goal 2

#### What was the objective you set yourself?

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

#### What actions did you take to achieve it?

First I read and understood the formula derived for cover times of simple random walks on a polygon. Since the random conductance model was more general, some methods could not be applied to it from the simple model. This required more careful analysis. However, the ideas of decomposing the chains and conditioning on the last visited vertex were similar. With the helps of my supervisor, the complicated parts were finally solved by a simple and clever intuition and Doob-h-transform. Though the final explicit formula was extreme complicated, it was indeed an explicit formula which could be calculated for given weights.

#### How do you know you have achieved your goal?

The terms constituting the cover time agree with the formula for the simple model, if we reduce our model to the simple model. Hence, the formula for our model is verified through the simple model.

The simulations showed that the cover times for some uniform random variables grows along with the formula for the simple model. And for some types of random variables, the cover times grows also in O(m^2), which agrees with the formula for the simple model.

#### What new or existing skills have you developed as a result of achieving this objective?

I learned to use Matlab to do symbolic calculations, analyse Markov chains in details and apply Doob-h-transform to find certain probabilities.

#### How will these support your research project, studies or career?

Complicated symbolic calculations can be annoying. Matlab helps to release me from the tedious work. This will save me lots of time if there are symbolic calculations in the future.

Markov chains is an important topic through my study. Hence a detailed analysis deepens my understanding.

Doob-h-transform is a method which can simplify the original problem significantly, which can be helpful in various situations. And there is also a continous version of it, which was applied in one of the talks of P@W workshop.

#### If you were to set yourself the same objective again, what would you do differently?

Since the method involving the Green's function failed, I either should examine it more carefully or study the Doob-h-transform in advance. However, these problems happened along the progress of project, which could be hardly foreseen.

## September 13, 2012

### Researcher Development Goal 1

#### What was the objective you set yourself?

Understanding the connections between random walks and electric networks.

#### What actions did you take to achieve it?

I read textbooks and papers which are related to this topic and tried to solve the similar questions in our extended model. I explained to my supervisor what I thought. My naive thinking about random walk was to break the chain directly to force the walk to go into certain directions. However, this was a too strong condition, since we were looking at walks that could go in both directions but happen to go certain one of them. This understanding was obtained through my discussion with my supervisor.

#### How do you know you have achieved your goal?

I successfully applied two key results about the connections between random walks and electric network in my project. One of them is about the probability of hitting certain state before another state, starting at a state between those two states. The other is about the expected times of visiting certain state before returning to the starting state.

These two results helped me to simplify the original problem, which were very useful.

#### What new or existing skills have you developed as a result of achieving this objective?

One of the skills is through reading papers to understand what the problem is and what the author tries to do. Papers are similar to lecture notes, but they are more compact and maybe refer to other sources. Hence they are more difficult and require more patience to read through them.

There are important connections between maths and physics. They may help each other to solve the problem. This project is a live example that renews my understanding about the interaction between physics and maths.

#### How will these support your research project, studies or career?

The ability to read papers is definitely an important skill in all research areas. It helps to know the new development of the subject and avoid the repeat of works. Also it gives me an example about how to present ideas clearly and precisely.

The understanding of connections between maths and physics may help me to solve difficult problems in the future, especially for providing ideas.

#### If you were to set yourself the same objective again, what would you do differently?

I think I probably should look at more complicated network and do some exercises. This is because the network in my project become complicated in the later stage. So a deep understanding would be more helpful.