Long range order for random field Ising and Potts models

Ding, Jian; Zhuang, Zijie

doi:10.1002/cpa.22127

Cited by 3 publications

(11 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This chapter follows the argument of [31] and proves phase transition for the nearest-neighbor Ising model with a random field. In Section 1 we present the model and the overall strategy of the Peierls' argument.…”

Section: Random Field Ising Modelmentioning

confidence: 74%

“…x∈A denotes the restriction of the external field to the subset A. The next Lemma was proved in [31] and is a direct consequence of Theorem 3.2.1. Lemma 3.2.4.…”

Section: 23)mentioning

confidence: 81%

“…The main idea used in Ding and Zhuang's proof of phase transition in [31] is to make the Peierls' argument on the joint space of the configurations and the external field, and when erasing a contour, perform in the external field the same flips you do in the configuration. Doing this, the part on the Hamiltonian that depends on the external field does not change, but the partition function does.…”

Section: Ding and Zhuang Approachmentioning

confidence: 99%

“…Recently, Ding and Zhuang [31], provided a simpler proof of the phase transition, not using RGM. In addition, Ding, Liu, and Xia [27] proved that if β c (d) is the critical inverse of the temperature of the Ising model with no field, for all β > β c (d) there exists a critical value ε 0 (d, β) such that the RFIM with ε ≤ ε 0 presents phase transition.…”

Section: Introductionmentioning

confidence: 99%

“…The argument from Ding and Zhuang in [31], for d ≥ 3, involves controlling the probability of a bad event, which is related to controlling the quantity sup…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range

Maia

View full text Add to dashboard Cite

To understand the meaning of the quotation opening this section, we need to know more about its author. Antônio Maia was a man with humble beginnings. Born in a village in the Northeast part of Brazil, his childhood was summarized by working in all sorts of plantations under the scorching sun of the semiarid climate of the region. At the time, in the 70's, this whole region of Brazil was very poor. In the city where he was born, for example, there were not any hospitals, schools, not even electric energy. At 18 years old, he moved to São Paulo, an emerging city at the time. Arriving here, he moved to a small house, living together with around 10 other people, relatives of his that had moved to the big city earlier. Living with no money in a city like São Paulo was no easy task, so he had to accept any kind of work available, from being a waiter, a construction worker, salesman. This is to say that most of the days of his life were a combination of hard work and tough decisions. Despite that, in the quote, he claimed he would gladly live all of those days as if they were his last. These words taught me to embrace the difficulties of life, to find happiness in them, and to not see them as the means to an end. This resonated with me throughout my Ph.D. As you may imagine, concluding a Ph.D. in mathematics, especially in a country like Brazil, is not an easy task. The frustrations and uncertainties of this path are tremendous, and so are the effort and the dedication needed to accomplish it. This beautiful idea of finding joy in the challenges of life was the key factor that helped me move forward, from undergrad student to a researcher, a scientist.The author of the quotation was my father, who tragically passed away a couple of months before my defense. To him, I own who I am.My family was the pillar that supported me all those years, and to them, I am profoundly grateful. To my mother Ivoneide Teixeira, who teaches me every day, through example, the importance of dedicating and sacrificing yourself to your work, a value I carry deeply in me. She, like my father, does not come from a privileged background and had to struggle and work hard all her life. Her i ii dedication to work and her unbreakable spirit, she never stops fighting for what she wants, are core values that I try to carry with me. She is the person I respect and own the most. I also thank my sisters, Fernanda Maia and Natália Maia, for the support they always gave me, for being an example to me in several ways, and for the joy you always brought to my life. It is a privilege to have you as my siblings.The last member of my family is my girlfriend, Jessica Dantas. We have been together since high school, so we have grown beside one another. Through the hardships and ups and downs of a relationship, I loved and admired her every step of the way. Her life was, in many ways, much tougher than mine. That never made her shine away from any challenge thrown at her. And after overcoming them, she still finds the will to pursue her future goals, which are ple...

show abstract

Section: Random Field Ising Modelmentioning

confidence: 74%

“…x∈A denotes the restriction of the external field to the subset A. The next Lemma was proved in [31] and is a direct consequence of Theorem 3.2.1. Lemma 3.2.4.…”

Section: 23)mentioning

confidence: 81%

Section: Ding and Zhuang Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The argument from Ding and Zhuang in [31], for d ≥ 3, involves controlling the probability of a bad event, which is related to controlling the quantity sup…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range

Maia

View full text Add to dashboard Cite

show abstract

Quantitative Disorder Effects in Low-Dimensional Spin Systems

Dario,

Harel,

Peled

2024

Commun. Math. Phys.

View full text Add to dashboard Cite

The Imry–Ma phenomenon, predicted in 1975 by Imry and Ma and rigorously established in 1989 by Aizenman and Wehr, states that first-order phase transitions of low-dimensional spin systems are ‘rounded’ by the addition of a quenched random field coupled to the quantity undergoing the transition. The phenomenon applies to a wide class of spin systems in dimensions $$d\le 2$$ d ≤ 2 and to spin systems possessing a continuous symmetry in dimensions $$d\le 4$$ d ≤ 4 . This work provides quantitative estimates for the Imry–Ma phenomenon: in a cubic domain of side length L, we study the effect of the boundary conditions on the spatial and thermal average of the quantity coupled to the random field. We show that the boundary effect diminishes at least as fast as an inverse power of $$\log \log L$$ log log L in general two-dimensional spin systems. For systems possessing a continuous symmetry, we show that the boundary effect diminishes at least as fast as an inverse power of L in two and three dimensions and at least as fast as an inverse power of $$\log \log L$$ log log L in four dimensions. Finally, we establish a partial uniqueness results for translation-covariant Gibbs states, and prove that, for almost every realization of the random field, all such states must agree on the thermally-averaged value of the quantity coupled to the random field. Specific models of interest for the obtained results include the random-field q-state Potts model, the Edwards-Anderson spin glass model, and the random-field spin O(n) models.

show abstract