All entries for Friday 14 September 2012
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.