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Phase Transitions in Ising models: the Semi-infinite with decaying field and the Random Field Long-range

João Maia

Abstract

In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field $h_i = λ|i_d|^{-δ}$, $λ$ is the wall influence, and $δ>0$. This external field decays as it gets further away from the wall. We are able to show that when $δ>1$ and $β> β_c(d)$, there exists a critical value $0< λ_c:=λ_c(δ,β)$ such that, for $λ<λ_c$ there is phase transition and for $λ>λ_c$ we have uniqueness of the Gibbs state. In addition, when $δ<1$ we have only one Gibbs state for any positive $β$ and $λ$. For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions $J_{xy}=|x-y|^{- α}$ when $α> d$ in dimension $d\geq 3$ if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions.

Phase Transitions in Ising models: the Semi-infinite with decaying field and the Random Field Long-range

Abstract

In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field , is the wall influence, and . This external field decays as it gets further away from the wall. We are able to show that when and , there exists a critical value such that, for there is phase transition and for we have uniqueness of the Gibbs state. In addition, when we have only one Gibbs state for any positive and . For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions when in dimension if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions.
Paper Structure (28 sections, 73 theorems, 477 equations, 19 figures)

This paper contains 28 sections, 73 theorems, 477 equations, 19 figures.

Table of Contents

  1. Semi-infinite Ising Model
  2. The model and important definitions
  3. Correlation inequalities and the limiting states
  4. The wetting transition with no field
  5. The macroscopic phenomenon of phase transition
  6. Low-temperature representation and Peierls contours
  7. The wetting transition in terms of contours
  8. Semi-infinite Ising model with inhomogeneous external fields
  9. External field decaying with l
  10. Critical behavior when d>1
  11. Uniqueness for d<1
  12. Random Cluster Representation and Edward-Sokal coupling
  13. The Edwards-Sokal model
  14. Uniqueness for $\delta$<1
  15. Random Field Ising Model
  16. ...and 13 more sections

Key Result

Theorem 1

Let $d\geq 2$, and let $J_c$ be the critical value of the Ising model in $\mathbb{Z}^d$ at $\beta=1$. Given any $\delta>0$, there exists a critical value $\overline{\lambda}_c=\overline{\lambda}_c(J,\delta)\geq 0$ such that the semi-infinite Ising model with external field $\widehat{\bm{\lambda}}$ p

Figures (19)

  • Figure 1: Complete wetting: $\lambda = 1 > \lambda_c$, $\beta=0.5$, $J=1$. The $+$ spins are black and the $-$ are white.
  • Figure 2: Partial wetting: $\lambda = 0.03 <\lambda_c$, $\beta=0.5$, $J=1$. The $+$ spins are black and the $-$ are white.
  • Figure 3: The - boundary condition of the semi-infinite model.
  • Figure 4: The influence of the external field in a box $\Lambda_n$.
  • Figure 5: The external field w.r.t. the distance of a spin to the wall.
  • ...and 14 more figures

Theorems & Definitions (143)

  • Theorem
  • Theorem
  • Proposition 1.2.1: GKS inequalities
  • Theorem 1.2.2: FKG Inequality
  • Lemma 1.2.3
  • Lemma 1.2.4
  • Lemma 1.2.5
  • Proposition 1.2.6
  • Theorem 1.2.7: Duplicate variable inequalities
  • Corollary 1.2.8
  • ...and 133 more