Texts on physics, maths and programming (562422), страница 25

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 25)

Loewner’s theorem tells us that , when expressed as a function of g_t (z), should be holomorphic in apart from a simple pole at e^iθ_t. Since g_t preserves the unit circle outside should be pure imaginary when |g_t (z)| = 1, and in order that , it should approach 1 as g_t (z) → 0. The only possibility is

This is the radial Loewner equation. In fact this is the version considered by Löewner [4].

It can now be argued, as before, that given lk and lk (suitably reworded to cover the case when r₂ is an interior point) together with reflection, θ_t must be proportional to a standard Brownian motion. This defines radial SLE. It is not immediately obvious how the radial and chordal versions are related. However, it can be shown that, if the trace of radial SLE hits the boundary of the unit disc at e^iθ_t1 at time t₁, then the law of K_t in radial SLE, for t < t₁, is the same chordal SLE conditioned to begin at e^iθ(0) and end at e^iθ_t1, up to a reparametrisation of time. This assures us that, in using the chordal and radial versions with the same κ, we are describing the same physical problem.

However, one feature that the trace of radial SLE possesses which chordal SLE does not is the property that it can wind around the origin. The winding angle at time t is simply θ_t − θ₀. Therefore, it is normally distributed with variance κt. At this point we can make a connection to the Coulomb gas analysis of the O (n) model in Section 2.4.1, where it was shown that the variance in the winding angle on a cylinder of length L is asymptotically (4/g)L. A semi-infinite cylinder, parametrised by w, is conformally equivalent to the unit disc by the mapping z = e^−w. Asymptotically, Re w → Re w − t under Loewner evolution. Thus, we can identify L t and hence

3.6.1. Identification with lattice models

This result allows use to make a tentative identification with the various realisations of the O (n) model described in Section 2.2. We have, using (27), n = −2 cos(4π/κ) with 2 κ 4 describing the critical point at x_c, and 4 < κ 8 corresponding to the dense phase. Some important special cases are therefore:

• κ = −2: loop-erased random walks (proven in [24]);

• κ = 8/3: self-avoiding walks, as already suggested by the restriction property, Section 3.5.2; unproven, but see [22] for many consequences;

• κ = 3: cluster boundaries in the Ising model, however, as yet unproven;

• κ = 4: BCSOS model of roughening transition (equivalent to the 4-state Potts model and the double dimer model), as yet unproven; also certain level lines of a gaussian random field and the ‘harmonic explorer’ (proven in [23]); also believed to be dual to the Kosterlitz–Thouless transition in the XY model;

• κ = 6: cluster boundaries in percolation (proven in [7]);

• κ = 8: dense phase of self-avoiding walks; boundaries of uniform spanning trees (proven in [24]).

It should be noted that no lattice candidates for κ > 8, or for the dual values κ < 2, have been proposed. This possibly has to do with the fact that, for κ > 8, the SLE trace is not reversible: the law on curves from r₁ to r₂ is not the same as the law obtained by interchanging the points. Evidently, curves in equilibrium lattice models should satisfy reversibility.

4. Calculating with SLE

SLE shows that the measure on the continuum limit of single curves in various lattice models is given in terms of one-dimensional Brownian motion. However, it is not at all clear how thereby to deduce interesting physical consequences. We first describe two relatively simple computations in two-dimensional percolation which can be done using SLE.

4.1. Schramm’s formula

Given a curve γ connecting two points r₁ and r₂ on the boundary of a domain , what is the probability that it passes to the left of a given interior point? This is not a question which is natural in conventional approaches to critical behaviour, but which is very simply answered within SLE [25].

As usual, we can consider to be the upper half plane, and take r₁ = a₀ and r₂ to be at infinity. The curve is then described by chordal SLE starting at a₀. Label the interior point by the complex number ζ.

Denote the probability that γ passes to the left of ζ by (we include the dependence on to emphasise the fact that this is a not a holomorphic function). Consider evolving the SLE for an infinitesimal time dt. The function g_dt will map the remainder of γ into its image γ′, which, however, by lk and lk, will have the same measure as SLE started from . At the same time, ζ → g_dt (ζ) = ζ + 2dt/(ζ − a₀). Moreover, γ′ lies to the left of ζ′ iff γ lies to the left of ζ. Therefore

where the average … is over all realisations of Brownian motion dB_t up to time dt. Taylor expanding, using dB_t = 0 and (dB_t)² = dt, and equating the coefficient of dt to zero gives

Thus, P satisfies a linear second-order partial differential equation, typical of conditional probabilities in stochastic differential equations.

By scale invariance P in fact depends only on the angle θ subtended between ζ − a₀ and the real axis. Thus, (29) reduces to an ordinary second-order linear differential equation, which is in fact hypergeometric. The boundary conditions are that P = 0 and 1 when θ = π and 0, respectively, which gives (specialising to κ = 6)

Note that this may also be written in terms of a single quadrature since one solution of (29) is P = const.

4.2. Crossing probability

Given a critical percolation problem inside a simply connected domain , what is the probability that a cluster connects two disjoint segments AB and CD of the boundary? This problem was conjectured to be conformally invariant and (probably) first studied numerically in [26]. A formula based on CFT as well as a certain amount of guesswork was conjectured in [1]. It was proved, for the continuum limit of site percolation on the triangular lattice, by Smirnov [7].

Within SLE, it takes a certain amount of ingenuity [5] to relate this problem to a question about a single curve. As usual, let be the upper half plane. It is always possible to make a fractional linear conformal mapping which takes AB into (−∞, x₁) and CD into (0, x₂), where x₁ < 0 and x₂ > 0. Now go back to the lattice picture and consider critical site percolation on the triangular lattice in the upper half plane, so that each site is independently coloured black or white with equal probabilities 1/2. Choose all the boundary sites on the positive real axis to be white, all those on the negative real axis to be black (see Fig. 8). There is a cluster boundary starting at the origin, which, in the continuum limit, will be described by SLE₆. Since κ > 4, it repeatedly hits the real axis, both to the L and R of the origin. Eventually every point on the real axis is swallowed. Either x₁ is swallowed before x₂, or vice versa.

(9K)

Fig. 8. Is there a crossing on the white discs from (0, x₂) to (−∞, x₁)? This happens if and only if x₁ gets swallowed by the SLE before x₂.

Note that every site on the L of the curve is black, and every site on its R is white. Suppose that x₁ is swallowed before x₂. Then, at the moment it is swallowed, there exists a continuous path on the white sites, just to the R of the curve, which connects (0, x₂) to the row just above (−∞, x₁). On the other hand, if x₂ is swallowed before x₁, there exists a continuous path on the black sites, just to the L of the curve, connecting 0− to a point on the real axis to the R of x₂. This path forms a barrier (as in the game of Hex) to the possibility of a white crossing from (0, x₂) to (−∞, x₁). Hence there is such a crossing if and only if x₁ is swallowed before x₂ by the SLE curve.

Recall that in Section 3.4.1 we related the swallowing of a point x₀ on the real axis to the vanishing of x_t = g_t (x_t) − a_t, which undergoes a Bessel process on the real line. Therefore

Denote this by P (x₁, x₂). By generalising the SLE to start at a₀ rather than 0, we can write a differential equation for this in similar manner to (29)

Translational invariance implies that we can replace ∂_a0 by -(∂_x1+∂_x2). Finally, P can in fact depend only on the ratio η = (x₂ − a₀)/(a₀ − x₁), which again reduces the equation to hypergeometric form. The solution is (specialising to κ = 6 for percolation)

It should be mentioned that this is but one of a number of percolation crossing formulae. Another, conjectured by Watts [27], for the probability that there is cluster which simultaneously connects AB to CD and BC to DA, has since been proved by Dubédat [28]. However, other formulae, for example for the mean number of distinct clusters connecting AB and CD [29], and for the probability that exactly N distinct clusters cross an annulus [30], are as yet unproven using SLE methods.

4.3. Critical exponents from SLE

Many of the critical exponents which have previously been conjectured by Coulomb gas or CFT methods may be derived rigorously using SLE, once the underlying postulates are assumed or proved. However SLE describes the measure on just a single curve, and in the papers of LSW a great deal of ingenuity has gone into showing how to relate this to all the other exponents. There is not space in this article to do these justice. Instead we describe three examples which give the flavour of the arguments, which initially may appear quite unconventional compared with the more traditional approaches.

4.3.1. The fractal dimension of SLE

Характеристики

Список файлов учебной работы