Percolation is a probabilistic model of wetting of porous medium, the spread of blight in an orchard, a forest fire, etc. Specifically, bond percolation is defined by giving the occupied and vacant states with probability
![](https://chart.googleapis.com/chart?cht=tx&chl=%24p%20D(y%20-%20x)%24&chf=bg,s,333333&chco=ffffff)
and
![](https://chart.googleapis.com/chart?cht=tx&chl=%241%20-%20p%20D(y%20-%20x)%24&chf=bg,s,333333&chco=ffffff)
, respectively, for each edge
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5C%7Bx%2C%20y%5C%7D%24&chf=bg,s,333333&chco=ffffff)
on a graph. It is known that phase transitions occur at a critical point
![](https://chart.googleapis.com/chart?cht=tx&chl=%24p_%5Cmathrm%7Bc%7D%24%20&chf=bg,s,333333&chco=ffffff)
in this model, and it is believed that some quantities exhibit power-law (critical phenomena). For example, it is predicted that the susceptibility (the mean cluster size)
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cchi_p%24&chf=bg,s,333333&chco=ffffff)
asymptotically behaves like
![](https://chart.googleapis.com/chart?cht=tx&chl=%24(p_%5Cmathrm%7Bc%7D%20-%20p)%5E%7B-%5Cgamma%7D%24&chf=bg,s,333333&chco=ffffff)
. The exponent
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5Cgamma%24&chf=bg,s,333333&chco=ffffff)
particularly takes the value
![](https://chart.googleapis.com/chart?cht=tx&chl=%241%24%20&chf=bg,s,333333&chco=ffffff)
in high dimension, which is called a mean-field value. In this talk, I will explain the basic topics of mean-field behavior for percolation models. I will also mention the infrared bound and the lace expansion. They are key topics in my research.