M.Sc Student | Greenberg Lev |
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Subject | Limit Properties of Random Fields |

Department | Department of Applied Mathematics |

Supervisor | Professor Dmitry Ioffe |

We consider a 2D Ising nearest neighbor model in a vertical strip of the width N at low temperatures, i.e., for all the temperatures where the Gibbs measure in the infinite volume is not unique. The positive boundary condition on the upper half of the strip and the negative boundary condition on the lower half of the strip impose a phase-separation line (+/- interface) to cross the strip.

Taking N to infinity we
obtain a sequence of the phase-separation lines. We show that there exists an
approximating random element in C[0,N], such that a Hausdorff distance between
the approximation and the phase-separation line in a vertical strip of the
width N is bounded by (log N)^{2}. Normalizing the approximation line
by N in the x-axis and by N^{1/2} in the y-axis we obtain a normalized approximation
line.

We prove that the sequence of the normalized approximating lines converges weakly to the Brownian bridge with the variance which is equal to the curvature of the Wulff shape.

In our analysis we represent the approximation line as a random walk. The uniform exponential mixing of the random walk enables us to switch to a Ruelle operator setup. Then we use standard probability techniques in order to prove the weak convergence.

The main goal of our work was to obtain the results mentioned above for the whole low temperature regime. First we obtain these results for the 2-point function representation model with free boundary conditions in the high temperature regime. Then by using the duality principle we receive the same results for the phase-separation lines in the low temperature regime.