## FUKUSHIMA, Ryoki

Name

**FUKUSHIMA, Ryoki**
Position
Associate Professor

E-Mail
ryoki (email address: add @kurims.kyoto-u.ac.jp)

Research

[Probability Theory]

I am working on stochastic processes on random media, such as the Anderson model, random walk in random environment and spin glasses.

The model I have been studying long is the so-called parabolic Anderson model. This is a model of diffusion particle or polymer chain which receives attractive or repulsive force from spatial impurities. My main contributions in this field are

1. Asymptotics of the partition function for the impurities given by a random perturbation of the lattice,

2. Localization of the polymer when the force from an impurity is of long range nature,

3. Localization of the polymer under the presence of small drift.

Another recent topic is a quantitative study of free energy of disordered statistical physics systems. Typically, the existence of free energy is proved by a certain ergodic theorem but this way makes it difficult to understand further properties. So far, I studied the free energy of the directed polymers in random environment and was able to prove the continuity of the free energy at zero temperature. The zero temperature limit appears not only in statistical physics but also in the combinatorial optimization. Thus it would be interesting to see to what extent the continuity can be justified theoretically.

I am working on stochastic processes on random media, such as the Anderson model, random walk in random environment and spin glasses.

The model I have been studying long is the so-called parabolic Anderson model. This is a model of diffusion particle or polymer chain which receives attractive or repulsive force from spatial impurities. My main contributions in this field are

1. Asymptotics of the partition function for the impurities given by a random perturbation of the lattice,

2. Localization of the polymer when the force from an impurity is of long range nature,

3. Localization of the polymer under the presence of small drift.

Another recent topic is a quantitative study of free energy of disordered statistical physics systems. Typically, the existence of free energy is proved by a certain ergodic theorem but this way makes it difficult to understand further properties. So far, I studied the free energy of the directed polymers in random environment and was able to prove the continuity of the free energy at zero temperature. The zero temperature limit appears not only in statistical physics but also in the combinatorial optimization. Thus it would be interesting to see to what extent the continuity can be justified theoretically.