## Critical Region for Droplet Formation in the Two-Dimensional Ising Model (2002)

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Citations: | 14 - 8 self |

### BibTeX

@MISC{Biskup02criticalregion,

author = {Marek Biskup and Lincoln Chayes and Roman Kotecky},

title = {Critical Region for Droplet Formation in the Two-Dimensional Ising Model},

year = {2002}

}

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### Abstract

We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L inverse temperature # > # c and overall magnetization conditioned to take the value m vL , where # -1 c is the critical temperature, m (#) is the spontaneous magnetization and vL is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v L L -2 tends to a definite limit. Specifically, we identify a dimensionless parameter #, proportional to this limit, a non-trivial critical value # c and a function ## such that the following holds: For # < # c , there are no droplets beyond log L scale, while for # > # c , there is a single, Wulff-shaped droplet containing a fraction ## # c = 2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, ## and # are related via a universal equation that apparently is independent of the details of the system.

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Citation Context ...tablished. We proceed by listing the properties of the two-dimensional model which will ultimately be needed in this paper. For general overviews of various aspects mentioned below we refer to, e.g., =-=[31, 54, 32, 14]-=-. The ΛDROPLET FORMATION IN THE 2D ISING MODEL 5 readers familiar with the background (and the standard notation) should feel free to skip the remainder of this section and go directly to Section 1.3... |

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Citation Context ...tablished. We proceed by listing the properties of the two-dimensional model which will ultimately be needed in this paper. For general overviews of various aspects mentioned below we refer to, e.g., =-=[31, 54, 32, 14]-=-. The ΛDROPLET FORMATION IN THE 2D ISING MODEL 5 readers familiar with the background (and the standard notation) should feel free to skip the remainder of this section and go directly to Section 1.3... |

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Citation Context ...for E I L follows. Similarly for EIII L . Proof of Lemma 3.2. We make liberal use of the correspondence between the graphical configurations ω and (sets of) spin configurations as described, e.g., in =-=[2,30,12]-=-. Each connected cluster in ω represents the spin configurations in which all sites of the cluster have spins of the same type. Thus, if EI L ∩ EII∗ L ∩ EIII L occurs, then the inner circuit of occupi... |

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Citation Context ...tablished. We proceed by listing the properties of the two-dimensional model which will ultimately be needed in this paper. For general overviews of various aspects mentioned below we refer to, e.g., =-=[31, 54, 32, 14]-=-. The ΛDROPLET FORMATION IN THE 2D ISING MODEL 5 readers familiar with the background (and the standard notation) should feel free to skip the remainder of this section and go directly to Section 1.3... |

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Citation Context ...thermodynamic limit, characterized by the magnetization m ⋆ (β) = 〈σ0〉 +,β , (1.3) marks the region of phase coexistence in this model. Indeed, there is a non-trivial critical value βc ∈ (0, ∞)—known =-=[47, 41, 1, 6]-=- to satisfy e2βc = 1 + √ 2—such that for β > βc, we have m⋆ (β) > 0 and there are multiple infinite-volume Gibbs states, while for β ≤ βc, the magnetization vanishes and there is a unique infinite-vol... |

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Citation Context ...of a minority phase. A mathematical proof of this fact starting from a system defined on the microscopic scale has been given in the context of percolation and Ising systems, first in dimension d = 2 =-=[4, 27]-=- and, more recently, in all dimensions d ≥ 3 [21, 13, 22]. Other topics related to the droplet shape have intensively been studied: Fluctuations of a contour line [18–20,26,3,37], wetting phenomena [5... |

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Citation Context ...two-dimensional (see [42] and references therein). Moreover, there are purported applications of Ising systems undergoing “fragmentation” in such diverse areas as nuclear physics and adatom formation =-=[36]-=-. From the perspective of statistical physics, perhaps more important are the investigations of small systems at parameter values corresponding to a first order4 M. BISKUP, L. CHAYES AND R. KOTECK ´Y... |

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Citation Context ... Further, using 〈A;B〉+,β to denote the truncated correlation function 〈AB〉+,β − 〈A〉+,β〈B〉+,β , the magnetic susceptibility, defined by χ(β) = ∑ 〈σ0;σx〉 +,β , (1.4) x∈Z 2 is finite for all β > βc, see =-=[24, 53]-=-. By the GHS or FKG inequalities, we have χ(β) ≥ 1 − m ⋆ (β) 2 > 0 for all β ∈ [0, ∞). • Peierls’ contours. Our next requisite item is a description of the Ising configurations in terms of Peierls’ co... |

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Citation Context ...for the Ising (and Potts) ferromagnets is by now a well established tool. The purpose of the following remarks is to define our notation; for more background and details we refer the reader to, e.g., =-=[35,12]-=- or the excellent review [32]. Let T ⊂ Z 2 denote a finite graph. A bond configuration, generically denoted by ω, is the assignment of a zero (vacant) or a one (occupied) to each bond in T. The weight... |

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Citation Context ...ct starting from a system defined on the microscopic scale has been given in the context of percolation and Ising systems, first in dimension d = 2 [4, 27] and, more recently, in all dimensions d ≥ 3 =-=[21, 13, 22]-=-. Other topics related to the droplet shape have intensively been studied: Fluctuations of a contour line [18–20,26,3,37], wetting phenomena [50] and Gaussian fields near a “wall” [5,15,29]. See [14] ... |

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Citation Context ... questions in the context of the Ising model at low temperatures. Subsequent developments [48, 49, 38, 39] have allowed the extension, in d = 2, of the aforementioned results up to the critical point =-=[40]-=-. Specifically, what has so far been shown is as follows: For two-dimensional volumes ΛL of side L and δ > 0 arbitrarily small, if the magnetization deficit exceeds L 4/3+δ , then a Wulff droplet acco... |

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Citation Context ...roof will use that, for any B ⊂ Z2 we have 0 ≤ 〈σx〉 +,β B +,β − 〈σx〉 B∪{y} ≤ e−‖x−y‖/ξ . (2.37) This inequality is a direct consequence of properties (1-2) above. The original derivation goes back to =-=[17]-=-. The bound (2.37) immediately implies both (2.34) and (2.35). Indeed, using (2.37) for all x ∈ A and y ∈ B \ A, we have for all A ⊆ B ⊆ Z2 that where α ′′ 0 ≤ 〈MA〉 +,β A 1 = α′′ 1 note that |MA − MA\... |

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Citation Context ... all x. Note that, via (2.33), the exponential decay (2.32) holds uniformly in A ⊂ Z 2 . Part (1) is a consequence of the main result of [24], see [53]; the GHS inequality from part (2) dates back to =-=[34]-=-. Now we are ready to state the desired estimates. Let A ⊂ Z2 be a finite set and let s be a scale function. Let P +,β,s A be the Gibbs measure of the Ising model in A ⊂ Z2 conditioned onthe event {Γ... |

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Citation Context ...t such possibilities it is necessary to demonstrate the absence of these “intermediate-sized” droplets and the insignificance—or absence—of large fluctuations. This was argued on a heuristic level in =-=[10]-=- and will be proven rigorously here. Thus, instead of blending into each other through a series of intermediate scales, the dropletdominated and the fluctuation-dominated regimes meet—literally—at a s... |

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Citation Context ...at parameter values corresponding to a first order4 M. BISKUP, L. CHAYES AND R. KOTECK ´Y transition in the bulk. In these situations, non-convexities appear in finite-volume thermodynamic functions =-=[36, 51, 44, 43]-=-, which naturally suggest the appearance of a droplet. Several papers have studied the disappearance of droplets and reported intriguing finite-size characteristics [52, 51, 45, 42, 9, 46, 7]. It is h... |

14 |
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Citation Context ...s d ≥ 3 [21, 13, 22]. Other topics related to the droplet shape have intensively been studied: Fluctuations of a contour line [18–20,26,3,37], wetting phenomena [50] and Gaussian fields near a “wall” =-=[5,15,29]-=-. See [14] for a summary of these results and comments on the (recent) history of these developments. The initial stages of the rigorous “Wulff construction” program have focused on systems in which t... |

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Citation Context ...for the Ising (and Potts) ferromagnets is by now a well established tool. The purpose of the following remarks is to define our notation; for more background and details we refer the reader to, e.g., =-=[35,12]-=- or the excellent review [32]. Let T ⊂ Z 2 denote a finite graph. A bond configuration, generically denoted by ω, is the assignment of a zero (vacant) or a one (occupied) to each bond in T. The weight... |

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Citation Context ...tood at a mathematically sophisticated level for many years. However, an analysis of systems at phase coexistence which contain droplets has begun only recently. Over a century ago, Curie [25], Gibbs =-=[33]-=- and Wulff [55] derived from surface-thermodynamical considerations that a single droplet of a particular shape—the Wulff shape—will appear in systems that are forced to exhibit a fixed excess of a mi... |

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Citation Context ...2.13) Wβ(S) = ∑ S∈S ( ) Wβ P(S) . (2.14) Proof. This is exactly Eq. (1.3.1) in [40]. The proof goes back to [48], Lemma 6.7. For our purposes, the key “splitting” argument is provided in Lemma 5.4 of =-=[49]-=-. A special case of the key estimate appears in Eq. (5.51) from Lemma 5.5 of [49] with the correct interpretation of the left-hand side. □ The bound (2.13) will be used in several ways: First, to show... |

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Citation Context ...he exponential costs of the two mechanisms, L 4/3 is the only conceivable scaling of the magnetization deficit where these are able to coexist. (This is the core of the heuristic approach outlined in =-=[9, 46]-=- and [7], see also [11, 8].) However, at the point where the droplets first appear, one can envision alternate scenarios involving complicated fluctuations and/or a multitude of droplets with effectiv... |

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Citation Context ...c (which corresponds to the high-temperature phase of the Ising system). Then, ( {0 ←→ ∂Λℓ} ) ≤ 4ℓe −ℓ/ξ (3.16) for all ℓ ≥ 1. P w,β ℓ,FK Proof. This is one portion of the proof of Proposition 4.1 in =-=[23]-=-. For the purposes of the next lemma, let n be a unit vector with rationally related components and let C(n) be the set of all pairs (a,b) of positive real numbers such that the a × b rectangle with s... |

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Citation Context ...he exponential costs of the two mechanisms, L 4/3 is the only conceivable scaling of the magnetization deficit where these are able to coexist. (This is the core of the heuristic approach outlined in =-=[9, 46]-=- and [7], see also [11, 8].) However, at the point where the droplets first appear, one can envision alternate scenarios involving complicated fluctuations and/or a multitude of droplets with effectiv... |

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Citation Context ...f how droplets disappear is by no means an esoteric issue. Aside from the traditional, i.e., three-dimensional, setting, there are experimental realizations which are effectively two-dimensional (see =-=[42]-=- and references therein). Moreover, there are purported applications of Ising systems undergoing “fragmentation” in such diverse areas as nuclear physics and adatom formation [36]. From the perspectiv... |

9 |
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Citation Context ...at parameter values corresponding to a first order4 M. BISKUP, L. CHAYES AND R. KOTECK ´Y transition in the bulk. In these situations, non-convexities appear in finite-volume thermodynamic functions =-=[36, 51, 44, 43]-=-, which naturally suggest the appearance of a droplet. Several papers have studied the disappearance of droplets and reported intriguing finite-size characteristics [52, 51, 45, 42, 9, 46, 7]. It is h... |

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Citation Context ...al costs of the two mechanisms, L 4/3 is the only conceivable scaling of the magnetization deficit where these are able to coexist. (This is the core of the heuristic approach outlined in [9, 46] and =-=[7]-=-, see also [11, 8].) However, at the point where the droplets first appear, one can envision alternate scenarios involving complicated fluctuations and/or a multitude of droplets with effective intera... |

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Citation Context ...thermodynamic limit, characterized by the magnetization m ⋆ (β) = 〈σ0〉 +,β , (1.3) marks the region of phase coexistence in this model. Indeed, there is a non-trivial critical value βc ∈ (0, ∞)—known =-=[47, 41, 1, 6]-=- to satisfy e2βc = 1 + √ 2—such that for β > βc, we have m⋆ (β) > 0 and there are multiple infinite-volume Gibbs states, while for β ≤ βc, the magnetization vanishes and there is a unique infinite-vol... |

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Citation Context ...7] and, more recently, in all dimensions d ≥ 3 [21, 13, 22]. Other topics related to the droplet shape have intensively been studied: Fluctuations of a contour line [18–20,26,3,37], wetting phenomena =-=[50]-=- and Gaussian fields near a “wall” [5,15,29]. See [14] for a summary of these results and comments on the (recent) history of these developments. The initial stages of the rigorous “Wulff construction... |

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Citation Context ...ied by the first bound in (4.9) for ∂Int instead of ∂Int◦ and the isoperimetric inequality |Λ| ≤ 1 16 |∂Λ|2 valid for any Λ ⊂ R2 that is a finite union of closed unit squares (see, e.g., Lemma A.1 in =-=[16]-=-). □ 4.2.3 Volume of large contours. The preceding lemma asserts that, for typical configurations, the interior of large contours is not too big. Actually, one can be a bit more precise. Namely, intro... |

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Citation Context ...odynamic functions [36, 51, 44, 43], which naturally suggest the appearance of a droplet. Several papers have studied the disappearance of droplets and reported intriguing finite-size characteristics =-=[52, 51, 45, 42, 9, 46, 7]-=-. It is hoped that the results established here will shed some light in these situations. 1.2 The model. The primary goal of this paper is a detailed description of the above droplet-formation phenome... |

5 |
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Citation Context ... been understood at a mathematically sophisticated level for many years. However, an analysis of systems at phase coexistence which contain droplets has begun only recently. Over a century ago, Curie =-=[25]-=-, Gibbs [33] and Wulff [55] derived from surface-thermodynamical considerations that a single droplet of a particular shape—the Wulff shape—will appear in systems that are forced to exhibit a fixed ex... |

5 |
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Citation Context ...s d ≥ 3 [21, 13, 22]. Other topics related to the droplet shape have intensively been studied: Fluctuations of a contour line [18–20,26,3,37], wetting phenomena [50] and Gaussian fields near a “wall” =-=[5,15,29]-=-. See [14] for a summary of these results and comments on the (recent) history of these developments. The initial stages of the rigorous “Wulff construction” program have focused on systems in which t... |