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Publication# Ising Model: Local Spin Correlations and Conformal Invariance

Abstract

We study the 2-dimensional Ising model at critical temperature on a simply connected subset of the square grid Z2. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of Hongler (Conformal invariance of Ising model correlations. Ph.D. thesis, [Hon10]), Hongler and Smirnov (Acta Math 211(2):191-225, [HoSm13]), Chelkak, Hongler, and Izyurov (Ann. Math. 181(3), 1087-1138, [CHI15]) to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.

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Related concepts (9)

Related publications (4)

Ising model

The Ising model (ˈiːzɪŋ) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors.

Analysis

Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development. The word comes from the Ancient Greek ἀνάλυσις (analysis, "a breaking-up" or "an untying;" from ana- "up, throughout" and lysis "a loosening"). From it also comes the word's plural, analyses.

Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.

This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions

In this thesis, we consider an anisotropic finite-range bond percolation model on $\mathbb{Z}^2$. On each horizontal layer $\{(x,i):x\in\mathbb{Z}\}$ for $i\in\mathbb{Z}$, we have edges $\langle(x,i),

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333-380, 1984). Both exhibit exactly solvable