Dynamics of the lattice spin systems

Phase transitions are familiar aspects of nature, which we experience in our daily life. One of the simplest example would be the liquid-solid transition of water. Ice and water are two different phases of the same substance, which appear as equilibrium states depending on the temperature. Presence of phase transition at 0^oC and its absence at, say, 10^oC can be compared as follows;

• At 0^oC, a block of ice stays as it is for a long time and so does a cup of water, since both of them are equilibrium states at 0^oC (Phase transition is, by definition, the existence of more than one equilibrium states).
• At 10^oC, both a block of ice and a cup of water, after a lapse of long enough time, will be water, since water is the unique equilibrium state at 10^oC.

The purpose of the current research program is to describe, as rigorous mathematical theorems, the nature of phase transitions in a way illustrated as above, i. e. by looking at the long time behavior of observables. We take lattice spin systems as the mathematical framework, one of them being the Ising model, whose physical back ground is quantum mechanical spinning of electrons.

References

1. (with K. S. Alexander) The spectral gap of 2-D stochastic Ising models with mixed boundary conditions,
   J. Stat. Phys. 104,, Nos. 1/2, 89--109, (2001).
   See the abstract. / Download the dvi file
2. The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice,
   Ann. Inst. Henri Poincar\'e. Probabilit\'es et Statistiques 37, 2, 223--243 (2001)
   See the abstract. / Download the dvi file
3. Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice,
   J. Funct. Anal. 173 , 74--102, (2000).
   See the abstract. / Download the dvi file
4. The log-Sobolev inequality for weakly coupled lattice field,
   Probab. Th. Rel. Fields , 115 , 1--40, (1999).
   See the abstract.
   Download the dvi file
5. Finite volume Glauber dynamics in a small magnetic field,
   J. Stat. Phys. 90 , Nos.3/4, (1998).
   See the abstract/ Download the dvi file
6. Exponential relaxation of finite volume Glauber dynamics near the border of the one phase region,
   in Trends in Probability and Related Analysis, the proceedings of SAP'96, ed. by N. Kono and N.-R. Shieh, World Scientific, 339--350, (1997).
   See the abstract/ Download the dvi file
7. Relaxed criteria of the Dobrushin-Shlosman mixing condition,
   J. Stat. Phys. 87, Nos.1/2, 293--309, (1997).
   See the abstract/ Download the dvi file
8. (with Schonmann, R.) Exponential relaxation of Glauber dynamics with some special boundary conditions,
   Commun. Math. Phys. 189, 299--310, (1997).
   See the abstract/ Download the dvi file
9. (with Higuchi, Y.) Slow relaxation of 2-D stochastic Ising models with random and non-random boundary conditions,
   In: New trends in stochastic analysis, ed. by K. D. Elworthy, S. Kusuoka, I. Shigekawa, World Scientific Publishing, 153--167, (1997).
   See the abstract/ Download the dvi file
10. (with Higuchi,Y.)Ising model on the lattice Sierpinski Gasket,
    J. Stat. Phys. , 84, Nos. 1/2 pp. 295--307, (1996).
    See the abstract/ Download the dvi file