Dipolar stochastic Loewner evolutions

Bauer, Michel; Bernard, Denis; Houdayer, J.

doi:10.1088/1742-5468/2005/03/p03001

Cited by 45 publications

(84 citation statements)

References 19 publications

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“…Suppose (Ω δ , B δ ) approximate (Ω, B), where B stands for +/ − / + / free boundary conditions. Then the probability that there is a + crossing between (b δ 2 a δ 2 ) and (a δ 1 b δ 1 ) (respectively, − crossing between (a δ 2 a δ 1 ) and (b δ 1 b δ 2 )) tends to 1 − G +/−/+/free (λ) (respectively, G +/−/+/free (λ)), where The formula (4.4) was conjectured in [BBK05] and proved in [Izy11], and (4.5) follows from the result of [HK13] and a calculation in [BBH05]. We do not know explicit formulae for other boundary conditions.…”

Section: Explicit Expressions For Observablesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

“…This approach, pursued in particular in [Smi01, LSW04, SS05, SS09, CN06, ChSm12, CDHKS13], was extremely fruitful. In a more general setup (e. g. for more complicated boundary conditions), the driving processes are typically described by Brownian motion B κt with time-dependent drifts; these processes do not admit such a simple axiomatic characterization anymore, and a lot of work has been done (see e. g. [LSW03,BBH05,BBK05,LK07,Dub07,Zha08,Dub09,IK13,FK13,KP14]) in order to understand them both in general and in relation to concrete lattice models.One celebrated result in the area is the proof of conformal invariance of fermionic observables in the critical Ising model [Smi06,ChSm12], leading in particular to the proof that interfaces in the model and its random cluster representation converge to SLE 3 and SLE 16 3 respectively [CDHKS13]. This result was extended in [Izy13] to radial and multiple SLE and to multiply-connected domains with suitable analogs of Dobrushin boundary conditions.…”

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confidence: 99%

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Smirnov’s Observable for Free Boundary Conditions, Interfaces and Crossing Probabilities

Izyurov¹

2015

Commun. Math. Phys.

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We prove convergence results for variants of Smirnov's fermionic observable in the critical planar Ising model in presence of free boundary conditions. One application of our analysis is a simple proof of a theorem by Hongler and Kytölä on convergence of critical Ising interfaces with plus-minus-free boundary conditions to dipolar SLE(3), and a generalization of this result to an arbitrary number of arcs carrying plus, minus or free boundary conditions. Another application is a computation of scaling limits of crossing probabilities in the critical FK-Ising model with arbitrary number of alternating wired/free boundary arcs. We also deduce a new crossing formula for the spin Ising model.The Stochastic Loewner evolution, introduced by Schramm in [Sch00], is a powerful tool in the study of lattice models in two-dimensional statistical mechanics at criticality. In this approach, one describes random geometric shapes arising in the models by planar growth processes. By means of Loewner's equation for the evolution of conformal maps, such processes can be encoded by continuous, real-valued "driving functions" (see, e. g., [Law05]).Schramm's original idea (often called "Schramm's principle") was that for certain boundary conditions, natural conformal invariance and "domain Markov property" assumptions on a random curve can be restated in terms of its driving function. In the scaling limit, these properties identify the latter as a Brownian motion B κt , where the intensity κ > 0 represents the universality class of the model. This approach, pursued in particular in [Smi01, LSW04, SS05, SS09, CN06, ChSm12, CDHKS13], was extremely fruitful. In a more general setup (e. g. for more complicated boundary conditions), the driving processes are typically described by Brownian motion B κt with time-dependent drifts; these processes do not admit such a simple axiomatic characterization anymore, and a lot of work has been done (see e. g. [LSW03,BBH05,BBK05,LK07,Dub07,Zha08,Dub09,IK13,FK13,KP14]) in order to understand them both in general and in relation to concrete lattice models.One celebrated result in the area is the proof of conformal invariance of fermionic observables in the critical Ising model [Smi06,ChSm12], leading in particular to the proof that interfaces in the model and its random cluster representation converge to SLE 3 and SLE 16 3 respectively [CDHKS13]. This result was extended in [Izy13] to radial and multiple SLE and to multiply-connected domains with suitable analogs of Dobrushin boundary conditions. Another very interesting case, namely that of free boundary conditions, was treated by Hongler and Kytölä [HK13], who proved a conjecture of [BBH05] that interfaces in the critical Ising model on a simply-connected domain with plus-minus-free boundary conditions converge to the dipolar SLE 3 , i. e., the SLE κ (ρ) process [LSW03, SW05] with κ = 3 and ρ = − 3 2 . The beautiful proof of Hongler and Kytölä was quite complicated, the main source of technical difficulties being that they did not use discrete...

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Section: Explicit Expressions For Observablesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Smirnov’s Observable for Free Boundary Conditions, Interfaces and Crossing Probabilities

Izyurov¹

2015

Commun. Math. Phys.

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“…To define dipolar SLE [12] one specifies a boundary point, which is going to be the starting point of the trace, and a boundary interval not including the starting point, which is going to included the termination point of the trace. Then dipolar SLE describes curves starting on the specified boundary point and stopped the first instant they hit the specified boundary interval.…”

Section: Dipolar Slementioning

confidence: 99%

2D growth processes: SLE and Loewner chains

Bauer¹,

Bernard²

2006

Physics Reports

195

357

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This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject. U, D = (planar) domain, ie. connected and simply connected open subset of the complex plane C. K = hulls, ie. connected compact subset of a domain D such that D \ K is a domain. Key words: PACS: * Member of CNRSγ [0,t] = the SLE curve with tip γ t at time t. K t = the SLE hull at time t. D t ≡ D \ K t , the domain D with the hull K t removed. g t : D t → D, the SLE Loewner map and f t : D → D t , its inverse. U t = g t (γ t )= image of the tip of the SLE curve. h t : D t → D, mapping the tip of the curve back to its starting point. vir = the Virasoro algebra. g h = a group element associated to a map h. G h = representation of g h in CFT Hilbert spaces. h r;s = [(rκ − 4s) 2 − (κ − 4) 2 ]/16κ for c = 1 − 6(κ − 4) 2 /4κ. |ψ r;s = highest weight vector with dimension h r;s . ϕ δ (x) = boundary primary field with dimension δ. ψ r;s (x) = degenerate boundary primary field with dimension h r;s . Φ(z,z) = bulk primary fields. 1 IntroductionThe main subject of this report is stochastic Loewner evolutions, and its interplay with statistical mechanics and conformal field theory.Stochastic Loewner evolutions are growth processes, and as such they fall in the more general category of growth phenomena. These are ubiquitous in the physical world at many scales, from crystals to plants to dunes and larger. They can be studied in many frameworks, deterministic of probabilistic, in discrete or continuous space and time. Understanding growth is usually a very difficult task. This is true even in two dimensions, the case we concentrate on in these notes. Yet two dimensions is a highly favorable situation because it allows to make use of the power of complex analysis in one variable. In many interesting cases, the growing object in two dimensions can be seen as a domain, i.e. a contractile open subset of the Riemann sphere (the complex plane with a point at infinity added) leading to a description by so-called Loewner chains.Stochastic Loewner evolution is a simple but particularly interesting example of growth process f...

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“…For κ ≥ 4, the probability that a dipolar SLE κ hull contains the bulk point z and the probability that it passes to the right of z can be computed (somewhat miraculously) again by routine martingale techniques by making the ansatz that they are harmonic functions of z and then checking that appropriate boundary conditions can be imposed, see again [10]. But the harmonic ansatz fails for κ < 4.…”

Section: An Elementary Application At κ =mentioning

confidence: 99%