MathOverview
Reputation
Next privilege 250 Rep.
 Jun 24 accepted Phase transition without the Peierls' counter argument Jun 24 comment Phase transition without the Peierls' counter argument But the computations due to Osager not also use the argument Peierls' argument? These are all possible proofs of the existence of a critical temperature and possibly the existence of a phase transition? Jun 24 revised Phase transition without the Peierls' counter argument added 4 characters in body Jun 24 asked Phase transition without the Peierls' counter argument May 11 comment 1 dimensional Ising model @Velenik I reviewed my previous conclusions. And really the term $\log 2$ is appropriate. I made a mistake. Or better two misconceptions. May 11 comment 1 dimensional Ising model @sreeram The calculations are correct. But I think the free energy would not be $f(\beta) = \lim_{L\to\infty} -\frac1{\beta L} \log Z_L^{+\varnothing} = - \frac1\beta(\log\cosh(\beta) + \log 2)$. The expression $\log 2$ is suspected.I think the Velenik would like you observe more carefully the calculations. May 3 comment Percolation and number of phases in the 2D Ising model See related question here. May 3 comment Percolation and number of phases in the 2D Ising model @Velenik I lack sufficient ability to handle the basic concepts to understand for myself the typical arguments of publications on statistical mechanics and especially on the Ising model. But I think I'll get the maturity that much hope in this area. May 3 comment Percolation and number of phases in the 2D Ising model @Velenik I will dedicate myself the next few weeks to chapter 6 of this book. I was very focused in the chapter on the Ising model. It is a very friendly text to a student. I believe this book will be a great reference in statistical mechanics. More and more I have good surprises with this book. I imagine that in the future will appear in the book, a section or chapter devoted to entropy and variational principles. I look forward to it. May 3 comment Percolation and number of phases in the 2D Ising model @Velenik It is much simpler than I thought. The result is set forth in equation (6.22) of the book (update on April 1). It is a direct consequence of the DLR equations and almost certain uniqueness of conditional probability. I was very excited about the motivations given to set the DLR equations. Especially with the discussion of why the Kolmogorov's extension theorem does not solve the problem of the existence of Gibbs measure. Thank you very much. May 2 comment Percolation and number of phases in the 2D Ising model By Gibbs measure in finite volume I want mean $\mu_{\Gamma}^\omega(A)=\sum_{\omega\in A}\frac{\exp\{\mathcal{H}_\Gamma^{-}(\omega)\}}{\mathcal{Z}_\Gamma^-}$ for all $A\in \{-1,+1\}^\Gamma$ and $\Gamma\subset \mathbb{Z}^2$. Here $\mathcal{H}_\Gamma^{-}(\omega)\}$ is the Hamiltonian. May 2 comment Percolation and number of phases in the 2D Ising model @Velenik Dear Velenik, it occurred to me a question some time that resists my efforts to answer it for myself. I was reluctant to ask here but... The question is as follows. Given any $\nu\in\mathcal{G}$ and set $\omega_i=-1\forall i \in\mathbb{Z}^2$ and $\mu_\Gamma^-(A)$ Gibbs measure in finite volume with boundary condition $-$. Is it true that $\nu(\cdot|\mathcal{F}_{\Gamma^c})(\omega)=\nu(\cdot|\mathcal{F}_{\Gamma^c})(-)=\‌​mu_\Gamma^-(\cdot)$ This equality seems to me that was used tacitly in resolution up several times. For example the strong Markov property and stochastic monotonicity. Mar 18 revised Percolation and number of phases in the 2D Ising model added 13 characters in body Sep 24 awarded Autobiographer Jul 2 awarded Curious Apr 26 revised Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities edited title Apr 25 comment Ising model. What is large fluctuations of magnetization? @YvanVelenik, Thank you. If my doubt is useful I will be happy to send it by email. Apr 19 comment Ising model. What is large fluctuations of magnetization? @hwlau Hello. Have some reference to this definition? Apr 19 revised Ising model. What is large fluctuations of magnetization? added 110 characters in body Apr 19 revised Ising model. What is large fluctuations of magnetization? edited body