LOGARITHMIC COEFFICIENTS AND GENERALIZED MULTIFRACTALITY OF WHOLE-PLANE SLE

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1 LOGARITHMIC COEFFICIENTS AND GENERALIZED MULTIFRACTALITY OF WHOLE-PLANE SLE BERTRAND DUPLANTIER 1), XUAN HIEU HO ), THANH BINH LE ), AND MICHEL ZINSMEISTER ) Abstract. It has been shown that for f an instance of the whole-plane SLE conformal map from the unit disk D to the slit plane, the derivative moments E f z) p ) can be written in a closed form for certain values of p depending continuously on the SLE parameter κ 0, ). We generalize this property to the mixed moments, E f z) ) p fz), along integrability curves in the moment q plane p, q) R depending continuously on κ, by extending the so-called Beliaev Smirnov equation to this case. The generalization of this integrability property to the m-fold transform of f is also given. We define a generalized integral means spectrum, βp, q; κ), corresponding to the singular behavior of the mixed moments above. The average generalized spectrum of wholeplane SLE takes four possible forms, separated by five phase transition lines in the moment plane R, whereas the average generalized spectrum of the m- fold whole-plane SLE is directly obtained from a linear map acting in that plane. We also conjecture the form of the universal generalized integral means spectrum. 1. Introduction 1.1. Logarithmic coefficients. Consider f, a holomorphic function in the unit disk D, 1) fz) = n 0 a n z n. Bieberbach observed in 1916 [4] that if f is further assumed to be injective, then a a 1, Date: May 8, 015. Key words and phrases. Whole-plane SLE, logarithmic moments, Beliaev Smirnov equation, generalized integral means spectrum, universal spectrum. The first author would like to thank the Isaac Newton Institute INI) for Mathematical Sciences, Cambridge, for its support and hospitality during the program on Random Geometry where part of this work was completed. He was partially supported by a fellowship from the Simons Foundation while at the INI. The research by the second author is supported by a joint scholarship from MENESR and Région Centre; the third author is supported by a scholarship of the Government of Vietnam. 1

2 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE t) f t 0 1 z t = f 1 t ) 8 t )= f t t) ) 0)= f 1) 0 f t 0)= 0 Figure 1. Loewner map z f t z) from D to the slit domain Ω t = C\γ[t, )) here slit by a single curve γ[t, )) for SLE κ 4 ). One has f t 0) = 0, t 0. At t = 0, the driving function λ0) = 1, so that the image of z = 1 is at the tip γ0) = f 0 1) of the curve. and he conjectured that a n n a 1 for all n >. This famous conjecture has been proved in 1984 by de Branges [5]. A crucial ingredient of his proof is the theory of growth processes that was developed by Loewner in 193 [], precisely in order to solve the n = 3 case of the Bieberbach conjecture. Let γ : [0, ) C be a simple curve such that γt) + as t + and such that γt) 0, t 0. Let Ω t = C \ γ[t, )) and f t : D = { z 1} Ω t be the Riemann map characterized by f t 0) = 0, f t0) > 0 See Fig. 1). It is easy to see that t f t0) is an increasing continuous function, diverging to + as t +. Assuming that f 00) = 1, and changing parameterization if necessary, we may set f t0) = e t, t 0. Loewner has shown that f t satisfies the following PDE: ) t f tz) = z z f tz) λt) + z λt) z, where λ : [0, ) D is a continuous function on the unit circle. With the sole information that λt) = 1, t, he could prove that a 3 3 a 1. Besides Loewner s theory of growth processes, de Branges proof also heavily relied on the considereration, developed by Grunsky [1] and later Lebedev and

3 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 3 Milin [17], of logarithmic coefficients. More precisely, if f : D C is holomorphic and injective with f0) = 0, we may consider the power series, 3) log fz) z = n 1 γ n z n. The purpose of introducing this logarithm was to prove Robertson s conjecture [6], which was known to imply Bieberbach s. Let f be in the class S of schlicht functions, i.e., holomorphic and injective in the unit disk, and normalized as f0) = 0, f 0) = 1. There is a branch f [] of z fz ) which is an odd function in S. Let us then write 4) f [] z) := z fz )/z = b n+1 z n+1, n=0 with b 1 = 1. Robertson s conjecture states that: n 5) n 0, b k+1 n + 1. k=0 The Lebedev and Milin approach to this conjecture consisted in observing that and consequently that log f [] z) z = 1 log fz) z, ) b n+1 z n = exp γ n z n. n=0 They proved what is now called the second Lebedev-Milin inequality, a combinatorial inequality connecting the coefficients of any power series to those of its exponential, namely 6) n 0, n b k+1 n + 1) exp k=0 n=1 1 n + 1 n m m=1 k=1 This naturally led Milin [4] to conjecture that n m 7) f S, n 1, k γ k 1 ) 0; k m=1 k=1 k γ k 1 k ) ). this conjecture, that de Branges proved in 1984, implies Robertson s, hence Bieberbach s conjecture. Returning to Loewner s theory, his derivation of Eq. ) above is only half of the story. There is indeed a converse: given any continuous function λ : [0, + [ C with λt) = 1 for t 0, the the Loewner equation ), supplemented by the boundary initial ) condition, lim t + f t e t z) = z, has a solution t, z)

4 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 4 f t z), such that f t z)) t 0 is a chain of Riemann maps onto simply connected domains Ω t ) that are increasing with t. In 1999, Schramm [8] introduced into the Loewner equation the random driving function, 8) λt) := κb t, where B t is standard one dimensional Brownian motion and κ a non-negative parameter, thereby making Eq. ) a stochastic PDE, and creating the celebrated Schramm-Loewner Evolution SLE κ. The associated conformal maps f t from D to C \ γ[t, ), obeying ) for 8), define the interior whole-plane Schramm-Loewner evolution. Their coefficients a n t), which are random variables, are defined by a normalized series expansion, as described in the following proposition [9]. Proposition 1.1. Let f t z)) t 0, z D, be the interior Schramm Loewner wholeplane process driven by λt) = e i κb t in Eq. ). We write 9) f t z) = e t z + n a n t)z n). and for its logarithm, 10) log e t f t z) z = n 1 γ n t)z n. Then the conjugate whole-plane Schramm Loewner evolution e i κb t f t e i κb t z ) has the same law as f 0 z), hence e in 1) κb t a n t) law) = a n 0). From this and Eqs. 9), 10), follows the identity e in κb t γ n t) law) = γ n 0). In the sequel, we set: a n := a n 0) and γ n := γ n 0). The starting point of the present article is the observation, made in Ref. [8], that the SLE κ process, in its interior whole-plane version, has a rich algebraic structure, giving rise to a host of integrability-like) closed form results. The first hint was the fact that, beyond the coefficient expectations Ea n ) for Eq. 9), the coefficient squared moments, E a n ), have very simple expressions for specific values of κ. This has been developed in detail in Refs. [9] and [18] see also [19, 0, 1]), by using the so-called Beliaev Smirnov equation, a PDE obeyed by the derivative moments E f z) p ), originally derived by the latter authors [3] to study the average integral means spectrum of the interior version of the) whole-plane SLE κ map. Note also that similar ideas already appeared in Ref. [13], where A. Kemppainen studied in detail the coefficients associated with the Schramm Loewner evolution, using a stationarity property of SLE [14]. However, the focus there was on expectations of the moments of those coefficients, rather than on the moments of their moduli.

5 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 5 Here, we study the logarithmic coefficients 10) of whole-plane SLE κ and the generalizations thereof, which are obtained by introducing generalized moments for the whole-plane SLE map, E f z) p / fz) q ), for p, q) R. The manifold identities so obtained in the p, q)-plane encompass all previous results. 1.. Main results. A first motivation of this article is the proof, originally obtained for small n by the third author [16], of the following. Theorem 1.1. Let fz) := f 0 z) be the time 0 unbounded whole-plane SLE κ map, in the same setting as in Proposition 1.1, such that then, for κ =, log fz) z = n 1 γ n z n ; E γ n ) = 1, n 1. n The idea behind the proof of Theorem 1.1 is to differentiate 3), and to compute E f z) fz) d fz) log dz z ). We indeed prove: = f z) fz) 1 z, Theorem 1.. Let f be the interior whole-plane SLE κ map, in the same setting as in Theorem 1.1; then for κ =, E z f z) ) 1 z)1 z) fz) =. 1 z z Let us briefly return to the Lebedev-Milin theory. By Theorem 1.1, we have for SLE, n m E k γ k 1 ) ) = 1 n m 1 k k = n + 1 n+1 1 k, m=1 k=1 m=1 k=1 k= which gives an example of the validity in expectation of the Milin conjecture. Recalling Definition 4), we also get, in expectation, a check of Robertson s conjecture 5): E log n k=0 b ) k+1 1 n + 1 n+1 Theorem 1. is actually a consequence of Theorems 3.1 and 4. of Sections 3 and 4 below, which give expressions in closed form for the mixed moments, f z)) p/ ) ) f z) p 11) a) E ; b) E, fz)) q/ fz) q k= 1 k.

6 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 6 along an integrability curve R, which is a parabola in the p, q) plane depending on the SLE parameter κ. In fact, we establish a general integrability result along the parabola R for the SLE two-point function: Gz 1, z ) := E z q 1 f z 1 )) p fz 1 )) q [ z q f z )) p fz )) q The mixed moments 11) can also be seen respectively as the value Gz, 0) of this SLE two-point function at z 1 = z, z 0) for a), and the value Gz, z) at coinciding points, z 1 = z = z, for b). These integrability theorems, which provide full generalizations of the results of Refs. [9] and [18], give rise to a host of new algebraic identities concerning the interior) whole-plane SLE κ random map. These integrability results can be generalized to the so-called m-fold symmetric transforms f [m], m N\{0}, of the whole-plane SLE map f. Interestingly enough, a linear map in the p, q)-moment plane allows one to directly relate the mixed moments of the m-fold map to those of f. The extension of the definition to m Z \ {0} exchanges exterior and interior whole-plane SLE maps, and relates their mixed moments; in particular, the case considered by Beliaev and Smirnov in Ref. [3] appears as the m = 1 transform of the interior SLE map studied here and in Ref. [9]. We thus believe that the general approach proposed here to consider generalized mixed moments in the p, q)-plane is the natural one for dealing with the properties of whole-plane SLE: in particular, it directly relates the inner and outer versions in a unified framework. Because of the introduction in Eq. 11) of the mixed moments b) of moduli of f and f for whole-plane SLE, it is also natural to define a generalized integral means spectrum βp, q), depending on p and q. It is associated with the possible singular behavior of circle integrals of such moments in D: ) f z) p E dz r 1 ) 1 r) βp,q), fz) q r D in the sense of the equivalence of the logarithms of both terms. In this article, we thus study the generalized spectrum, βp, q; κ), of wholeplane SLE κ in the whole parameter space p, q) R. We show that it takes four possible forms, β 0 p), β tip p), β lin p) and β 1 p, q). The first three spectra are independent of q, and are respectively given by the bulk, the tip and the linear SLE spectra appearing in the work by Beliaev and Smirnov [3] and for the bulk case, corresponding to the harmonic measure multifractal spectrum derived earlier by the first author in Ref. [6]). The fourth spectrum, β 1 p, q), is the extension to non-vanishing q of a novel integral means spectrum, which was discovered and studied in Refs. [9] and [18, 0], and which is due to the unboundedness of whole-plane SLE. As shown in Ref. [9], this spectrum is also closely related to the SLE tip exponents obtained by quantum gravity techniques in Ref. [7], and to the so-called radial SLE derivative exponents of Ref. [15]. Five different ]).

7 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 7 phase transition lines then appear to partition the p, q)-plane in four different domains, whose precise form is given. The generalization of this four-domain structure to the generalized integral means spectrum of m-fold transforms, f [m], m Z\{0}, is obtained in a straightforward way from the above mentioned linear map in p, q) co-ordinates. This structure appears so robust that the universal generalized spectrum Bp, q), i.e., the maximum of βp, q) over all unbounded univalent functions in the unit disk, presents a similar partition of the mixed moment plane. We give a precise conjecture for its four forms, which incorporates known results on the standard universal spectra for univalent, unbounded, functions [11, 5] Synopsis. This article is organized as follows. Section deals with logarithmic coefficients and with the proof of Theorems 1.1 and 1.. It sets up the martingale techniques needed for dealing with mixed moments. Section 3 uses them for the study of the complex one-point function a) in 11), which is shown to obey a simple differential equation in complex variable z. This leads to Theorem 3.1, which establishes a closed form for this function along the integrability parabola R in the p, q)-plane. Section 4 is concerned with the moduli one-point function b) in 11), and more generally, with the SLE two-point function Gz 1, z ). A PDE in z 1, z ) is derived for Gz 1, z ), which yields a proof of Theorem 4. establishing closed form expressions for G for all p, q) R. Section 5 deals with the generalization of the previous integrability results to the m-fold symmetric transforms f [m], m Z \ {0}, of the whole-plane SLE map f. Section 6 is devoted to the study of the averaged generalized spectrum βp, q; κ) of the whole-plane SLE κ random map f, as well as to the averaged generalized integral means spectrum β [m] p, q; κ) of the m-fold transform f [m] for m Z \ {0}. Of particular interest are the five phase transition lines separating the four different analytic expressions of β or β [m] ) in the moment plane. A more geometric derivation of these transition lines, based on a conic representation of the spectra, is given in Section 7. In the final Section 8, we give a full description of the expected form for the universal generalized integral means spectrum, Bp, q), in terms of known or conjectured results on the standard universal spectrum for univalent functions. Acknowledgments. It is a pleasure to thank Kari Astala for extended discussions about the universal generalized integral means spectrum.. Expectations of logarithmic coefficients.1. A martingale computation. In this section, we first prove the following: Theorem.1. Let fz) be the whole-plane SLE κ map, in the same setting as in Theorem 1.1; then for κ =, { 1/, n = 1, Eγ n ) = 0, n.

8 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 8 Differentiating both sides of 3), we get 1) z f z) fz) = 1 + nγ n z n. n 1 Let us now consider 13) Gz) := E z f ) z), fz) and, following Ref [3], aim at finding a partial differential equation satisfied by G. For the benefit of the reader not familiar with Ref. [3], let us detail the strategy of that paper that we will apply in various contexts here. The starting point is to consider the radial SLE κ, solution to the ODE t g t z) = g t z) λt) + g tz) λt) g t z), z D, with the initial condition g 0 z) = z, and where λt) = e i κb t. The map g t conformally maps a subdomain of the unit disk onto the latter. As we shall see shortly, the whole-plane map f is rather related to the map gt 1, but this last function satisfies, by Loewner s theory, a PDE not well-suited to Itô calculus. To overcome this difficulty, one runs backward the ODE of radial SLE, i.e., one compares gt 1 to g t. This is the purpose of Lemma 1 in [3] an analog of Lemma 3.1 in Ref. [7]), which states that, for t R, g t z) has the same law as the process f t z), defined as follows. Definition.. The conjugate, inverse) radial SLE process f t is defined, for t R, as 14) ft z) := gt 1 zλt))/λt). The lemma then results from the simple observation that f s z) = ĝ s z), where, for fixed s R, the new process ĝ t z) := g s+t gs 1 zλs))/λs) can be shown to be a radial SLE. This lemma implies in particular that f t is solution to the ODE: 15) t ft z) = f t z) f t z) + λt) f t z) λt), f0 z) = z. To apply Itô s stochastic calculus, one then uses Lemma in Ref. [3], which is a version of the SLE s Markov property, f t z) = λs) f t s f s z)/λs)). To finish, one has to relate the whole-plane SLE to the modified) radial one. This is done through Lemma 3 in [3], which is in our present setting with a change of an e t convergence factor there to an e t factor here, when passing from the exterior to the interior of the unit disk D):

9 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 9 Lemma.3. The limit in law, lim t + e t ft z), exists, and has the same law as the time zero) interior whole-plane random map f 0 z): lim t + et ft z) law) = f 0 z). Let us now turn to the proof of Theorem.1. Proof. Let us introduce the auxiliary, time-dependent, radial variant of the SLE one-point function Gz) 13) above, Gz, t) := E z f ) tz) 16), f t z) where f t is a modified radial SLE map at time t as in Definition.. Owing to Lemma.3), we have 17) lim Gz, t) = Gz). t + We then use a martingale technique ) to obtain an equation satisfied by Gz, t). f For s t, define M s := E t z) f F s, where F s is the σ-algebra generated by tz) {B u, u s}. M s ) s 0 is by construction a martingale. Because of the Markov property of SLE, we have [3] ) f M s = E t z) f t z) F s f = E s z) λs) = f sz) λs) E = f sz) Gz s, τ), f s z) f t s f ) s z)/λs)) f t s f s z)/λs)) F s f t s f ) s z)/λs)) f t s f s z)/λs)) F s where z s := f s z)/λs), and τ := t s. We have from Eq. 15) [ ] fs+λs) s log f z fs s f s λs) = = f s + λs) f s f s λs) λs) f s 18) f s λs)) = 1 1 z s ), 19) 0) s log f s = f s s = z s + 1 f s z s 1, dz s = z s [ zz + 1 z s 1 κ ] ds iz s κdbs.

10 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 10 The coefficient of the ds-drift term of the Itô derivative of M s is obtained from the above as, [ f 1) sz) z s f s z) 1 z s ) + z zs + 1 s z s 1 κ ) z τ κ ] z s z Gz s, τ), and vanishes by the local) martingale property. Because f s is univalent, f s does not vanish in D, therefore the bracket above vanishes. Owing to the existence of the limit 17), we can now take the τ + limit in the above, and obtain the ODE, P )[Gz)] := z z z) Gz) + z z 1 κ ) ) G z) κ z G z) [ = z ) z z) + z z κ ] z 1 z z) Gz) = 0. Following Ref. [9], we now look for solutions to Eq. ) of the form ϕ α z) := 1 z) α. We have P )[ϕ α ] = A,, α)ϕ α + B, α)ϕ α 1 + C, α)ϕ α, where, in anticipation of the notation that will be introduced in Section 3 below, A,, α) := α κ α, B, α) := 3 + κ ) α + κα, C, α) := + + κ ) α κ α, with, identically, A + B + C = 0. The linear independence of ϕ α, ϕ α 1, ϕ α thus shows that P )[ϕ α ] = 0 is equivalent to A = B = C = 0, which yields κ =, α = 1, and Gz) = 1 z. From Definition 13), we thus get Lemma.4. Let fz) = f 0 z) be the interior whole-plane SLE map at time 0, in the same setting as in Proposition 1.1; we then have E z f ) z) = 1 z. fz) Theorem.1 follows from Lemma.4 and the series expansion 1)... Proof of Theorem 1.1. Using 1), we get 3) z f z) fz) = 1 + nγ n z n + z n ) + nmγ n γ m z n z m. n 1 n 1 m 1 On the other hand, by Theorem 1., E z f z) ) 1 z)1 z) fz) = = 1 z n+1 z n z n z n+1 + z n z n. 1 z z) n 0 n 0 n 1

11 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 11 Identifying the latter with the expectation of 3), we get the expected coefficients Eγ 1 ) = 1/, Eγ n ) = 0, n, E γ n ) = 1 n, n 1, 1 Eγ n γ n+1 ) = nn + 1), Eγ n γ n+k ) = 0, n 1, k, which encompasses Theorems 1.1 and SLE one-point Function q p - p ) Figure. Integral curves R of Theorem 3.1, for κ = blue), κ = 4 red), and κ = 6 green). In addition to the origin, the q = 0 intersection point with the p-axis is at pκ) := 6 + κ) + κ)/8κ, with p) = p6) = [9, 18]. Let us now turn to the natural generalization of Lemma.4. Theorem 3.1. Let fz) = f 0 z) be the interior whole-plane SLE κ map at time zero, in the same setting as in Proposition 1.1. Consider the curve R, defined parametrically by p = κ γ + + κ ) γ, p q = 1 + κ ) 4) γ, γ R. On R, the whole-plane SLE κ one-point function has the integrable form, ) f z)) p E = 1 z) γ. fz)/z) q

12 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 1 Remark 3.. Eq. 4) describes a parabola in the p, q) plane see Fig. ), which is given in Cartesian coordinates by ) p q 5) κ 4 + κ) p q + κ + κ + p = 0, with two branches, γ = γ 0 ± p) := κ ± ) κ) κ κ) 8κp, p, 6) 8κ q = p 1 + κ ) γ 0 ± p). or, equivalently, 7) p = q + + κ 6 + κ ± ) κ) 8κ κ) 16κq, q 16κ. Proof. Our aim is to derive an ODE satisfied by the whole-plane SLE one-point function, ) 8) Gz) := E z q f z)) p, fz)) q which, by construction, stays finite at the origin and such that G0) = 1. Let us introduce the shorthand notation, 9) X t z) := f tz)) p f, t z)) q where f t is the conjugate, reversed radial SLE process in D, as introduced in Definition., and such that by Lemma.3, the limit, lim t + e t ft z) law) = f 0 z), is the same in law as the whole-plane map at time zero. Applying the same method as in the previous section, we consider the time-dependent function ) 30) Gz, t) := E z q Xt z), such that ) p q 31) lim exp t + t Gz, t) = Gz). Consider now the martingale M s ) t s 0, defined by M s = EX t z) F s ). By the SLE Markov property we get, setting z s := f s z)/λs), 3) M s = X s z) Gz s, τ), τ := t s. As before, the partial differential equation satisfied by Gz s, τ) is obtained by expressing the fact that the ds-drift term of the Itô differential of Eq. 3), dm s = G dx s + X s d G,

13 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 13 vanishes. The differential of X s is simply computed from Eqs. 18) and 19) above as: 33) dx s z) = X s z)f z s )ds, F z) := p [ ] 1 1 z) q [ 1 ]. 1 z The Itô differential d G brings in the ds terms proportional to zs G, zs G, and τ G; therefore, in the PDE satisfied by G, the latter terms are exactly the same as in the PDE 1). We therefore directly arrive at the vanishing condition of the overall drift term coefficient in dm s, [ 34) X s z) F z s ) + z s zs + 1 z s 1 κ ) z τ κ z s z ] Gz s, τ) = 0. Since X s z) does not vanish in D, the bracket in 34) must identically vanish: [ z s ) F z s ) + z s z s 1 z τ κ ] z s z ) Gz s, τ) = 0, where we used z z + z z = z z ). To derive the ODE satisfied by Gz) 8), we first recall its expression as the limit 30), which further implies lim exp τ + p q τ ) τ Gz, τ) = p q Gz). Multiplying the PDE 34) satisfied by G by exp p q τ) and letting τ +, we get [ P )[Gz)] := κ z z) 1 + z 1 z z z + F z) + p q ] Gz) [ = κ z z) 1 + z 1 z z p z 1 z) + q ] 36) 1 z + p q Gz) = 0. We now look specifically for solutions to 36), together with the boundary condition G0) = 1, of the form ϕ α z) = 1 z) α. This function satisfies the simple differential operator algebra [9] 37) P )[ϕ α ] = Ap, q, α)ϕ α + Bq, α)ϕ α 1 + Cp, α)ϕ α, where 38) 39) 40) Ap, q, α) := p q + α κ α, Bq, α) := q 3 + κ ) α + κα, Cp, α) := p + + κ ) α κ α,

14 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 14 such that, identically, A + B + C = 0. Because ϕ α, ϕ ϕ 1, ϕ α are linearly independent, the condition P )[ϕ γ ] is equivalent to the system A = C = 0, hence Cp, γ) = 0 and Ap, q, γ) Cp, γ) = p q 1 + κ/)γ = 0. It yields precisely the parabola parametrization 4) given in Theorem 3.1, and has for solution 6). 4. SLE two-point function 4.1. Beliaev Smirnov type equations. In this section, we will determine the mixed moments of moduli, E f z) ), p for p, q) belonging to the same parabola fz) q R as in Theorem 3.1, and where f = f 0 is the time zero) interior whole-plane SLE κ map. In contradistinction to the method used in Refs. [3, 9] for writing a PDE obeyed by E f z) p ), we shall use here a slightly different approach, building on the results obtained in Section.1. We shall study the SLE two-point function for z 1, z D, [ ]) 41) Gz 1, z ) := E z q f z 1 )) p 1 z q f z fz 1 )) q )) p. fz )) q As before, we define a time-dependent, auxiliary two-point function, Gz 1, z, t) := E z q f [ tz 1 )) p 1 f z q f ] t z 1 )) q tz )) p f t z )) q 4) ) = E z q 1 X t z 1 )z q X t z ), where as above f t is the reverse radial SLE κ process., and where we used the shorthand notation 9). This time, the two-point function 41) is the limit 43) lim t + ep q)t Gz1, z, t) = Gz 1, z ). Let us define the two-point martingale M s ) t s 0, with By the Markov property of SLE, M s := EX t z 1 )X t z ) F s ). 44) E X t z 1 )X t z ) F s ) = Xs z 1 )X s z ) Gz 1s, z s, τ), τ := t s, where 45) z 1s := f s z 1 )/λs); z s := f s z )/λs) = f s z )λs).

15 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 15 Their Itô differentials, dz 1s and d z s, are as in 0), [ z1s + 1 dz 1s = z 1s z 1s 1 κ ] ds i κ z 1s db s, 46) [ zs + 1 d z s = z s z s 1 κ ] ds + i κ z s db s. As before, the partial differential equation satisfied by Gz 1s, z s, τ) is obtained by expressing the fact that the ds-drift term of the Itô differential of Eq. 44), 47) dm s = [dx s z 1 )X s z ) + X s z 1 )dx s z )] G + X s z 1 )X s z ) d G, vanishes. The differentials of X s, X s are as in Eq. 33) above: 48) dx s z 1 ) = X s z 1 )F z 1s )ds, dx s z ) = X s z )F z s )ds, F z) := p q p 1 z) + q 1 z. We thus obtain the simple expression [ 49) dm s = X s z 1 )X s z ) [F z 1s ) + F z s )] G ds + d G ], and the vanishing of the ds-drift term in dm s requires that of the drift term in the right-hand side bracket in 49), since X s z) does not vanish in D. The Itô differential of Gz1s, z s, τ) can be obtained from Eqs. 46) and Itô calculus as 50) d Gz 1s, z s, τ) = 1 G dz1s + G d zs τ G ds κ z 1s 1 G ds κ z s G ds + κz 1s z s 1 G ds, where use was made of the shorthand notations, 1 := z1 and := z. We observe that the only coupling between the z 1s, z s variables arises in the last term of 50), the other terms simply resulting from the independent contributions of the z 1s and z s parts. Using again the Itô differentials 46), we can rewrite 50) as 51) d G = i κ z 1s 1 z s ) G dbs + z 1s + 1 z 1s 1 z 1s 1 G ds + z s + 1 z s 1 z s G ds τ G ds κ z 1s 1 z s ) G ds, where we used the obvious formal identity 5) z 1 1 ) + z ) z 1 1 z = z 1 1 z ). At this stage, comparing the computations 49) and 51) above with those in the one-point martingale study in Section.1, it is clear that the PDE obeyed by

16 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 16 G = Gz 1s, z s, τ) is obtained as two duplicates of Eq. 35), completed as in 5) by the derivative coupling between variables z 1s, z s : [ z 1s ) F z 1s )+z 1s z 1s 1 z s F z s )+ z s z s 1 τ κ ] z 1s 1 z s ) G = 0. The existence of the limit 43) further implies that of lim τ ep q)τ τ Gz1, z, τ) = p q)gz 1, z ). Multiplying the PDE 53) satisfied by G by expp q)τ) and letting τ +, then gives the expected PDE for Gz 1, z ). It can be most compactly written in terms of the ODE 36) as [ ] 54) P 1 ) + P ) + κz 1 1 z Gz1, z ) = 0, and its fully explicit expression is 55) PD)[Gz 1, z )] = κ z 1 1 z ) G 1 + z 1 z 1 1 G 1 + z z G 1 z 1 1 z [ p + 1 z 1 ) p 1 z ) + q + q ] + p q G = 0. 1 z 1 1 z 4.. Moduli one-point function. Note that one can take the z 1 = z = z case in Definition 41) above, thereby obtaining the moduli one-point function, 56) Gz, z) = E z q f ) z) p. fz) q Because of Eq. 55), it obeys the corresponding ODE, 57) PD)[Gz, z)] = κ z z ) G 1 + z 1 + z z G z G [ 1 z 1 z p + 1 z) p 1 z) + q 1 z + q ] + p q G = 0, 1 z which is the generalization to q 0 of the Beliaev Smirnov equation studied in Refs. [9] and [18] Integrable case. Lemma 4.1. The space of formal series F z 1, z ) = k,l N a k,lz k 1 z l, with complex coefficients and that are solutions of the PDE 55), is one-dimensional. Proof. We assume that F is a solution to 55) with F 0, 0) = 0; it suffices to prove that, necessarily, F = 0. We argue by contradiction: If not, consider the minimal necessarily non constant) term a k,l z k z l in the series of F, with a k,l 0 and k + l minimal and [ non vanishing). Then PD)[F ] 55) will have a minimal term, equal to a κ k,l k l) + k + l ] z1 k z, l which is non-zero, contradicting the fact that PD)[F ] vanishes.

17 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 17 As a second step, following Ref. [9], let us consider the action of the operator PD) of 55) on a function of the factorized form ϕz 1 )ϕ z )P z 1, z ), which we write, in a shorthand notation, as ϕ ϕp. By Leibniz s rule, it is given by PD)[ϕ ϕp ] = κ ϕ ϕz 1 1 z ) P κz 1 1 z )ϕ ϕ)z 1 1 z )P + κz 1 1 ϕ) z ϕ)p ϕ ϕ 1 + z 1 z 1 1 P ϕ ϕ 1 + z z P 1 z 1 1 z [ κ ϕz 1 1 ) ϕ + κ ϕ z ) ϕ + ϕ 1 + z 1 z 1 1 ϕ + ϕ 1 + z ] z ϕ P 1 z 1 1 z [ p + 1 z 1 ) p 1 z ) + q + q ] + p q ϕ ϕp. 1 z 1 1 z Note that the operator z 1 1 z is antisymmetric with respect to z 1, z ; therefore, if we choose a symmetric function, P z 1, z ) = P z 1 z ), the first line of PD)[ϕ ϕp ] above identically vanishes. One then looks for solutions to 55) of the particular form, Gz 1, z ) = ϕ α z 1 )ϕ α z )P z 1 z ), where, as before, ϕ α z) = 1 z) α. The action of the differential operator then takes the simple form, PD)[ϕ α ϕ α P ] =z 1 z ϕ α 1 ϕ α 1 κα P 1 z 1 z )P ) + P 1 )[ϕ α ] ϕ α P + P )[ ϕ α ]ϕ α P, where P is the derivative of P with respect to z 1 z, and P ) is the so-called boundary operator 36) [9]. The ODE, κα P x) 1 x)p x) = 0 with x = z 1 z and P 0) = 1, has for solution P z 1 z ) = 1 z 1 z ) κα /. It is then sufficient to pick for α the value γ = γ ± 0 p) 6) such that P )[ϕ γ ] = 0, as obtained in the proof of Theorem 3.1, to get a solution of the PDE, PD)[ϕ γ ϕ γ P ] = 0 55). By uniqueness of the solution with G0, 0) = 1, it gives the explicit form of the SLE two-point function, We thus get: Gz 1, z ) = ϕ γ z 1 )ϕ γ z )1 z 1 z ) κγ /. Theorem 4.. Let fz) = f 0 z) be the interior whole-plane SLE κ map in the setting of Proposition 1.1); then, for p, q) belonging to the parabola R defined in Theorem 3.1 by Eqs. 4) or 5) or 6), and for any pair z 1, z ) D D, E z q 1 f z 1 )) p fz 1 )) q [ z q f z )) p fz )) q ]) = 1 z 1) γ 1 z ) γ 1 z 1 z ) β, β = κ γ. Corollary 4.3. In the same setting as in Theorem 4., we have for z D, E z q f ) z) p = 1 z)γ 1 z) γ, β = κ fz) q 1 z z) β γ,

18 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 18 for γ = γ 0 ± p) := κ ± ) 4 + κ) κ 8κp, p q = p 1 + κ ) γ 0 ± p). 4 + κ), 8κ Let us stress some particular cases of interest. First, the p = 0 case gives some integral means of f. Corollary 4.4. The interior whole-plane SLE κ map has the integrable moments [ ] +κ)4+κ) [ ] +κ)4+κ) 4κ E fz1 ) 4κ fz ) = 1 z 1) 4+κ κ 1 z ) 4+κ κ, z 1 z 1 z 1 z ) 4+κ) κ fz) E z +κ)4+κ) ) κ = 4+κ 1 z) κ 1 z) 4+κ κ 1 z z) 4+κ) κ Second, taking p = q yields the logarithmic integral means we started with: Corollary 4.5. The interior whole-plane SLE κ map fz) = f 0 z) has the integrable logarithmic moment [ ] f +κ [ ] +κ z 1 ) κ f E z 1 z κ ) z ) = 1 z 1) κ 1 z ) κ, fz 1 ) fz ) 1 z 1 z ) κ E z f +κ z) ) κ fz) = 1 z) κ 1 z) κ. 1 z z) κ Theorem 1. describes the κ = case of the latter result. 5. Generalization to processes with m-fold symmetry The results of Section 4 may be generalized to functions with m-fold symmetry, with m a positive integer, as was studied in [9]. For f in class S, f [m] z) is defined as being the holomorphic branch of fz m ) 1/m whose derivative is equal to 1 at 0. These are the functions in S whose Taylor series is of the form fz) = k 0 a mk+1z mk+1. The m = case corresponds to odd functions that play a crucial role in the theory of univalent functions. One can also extend this definition to negative integers m, by considering then the m-fold transform of the outer whole-plane SLE as the conjugate by the inversion z 1/z of the m)-fold transform of the inner whole-plane SLE: f [m] z) = 1/f [ m] 1/z) for m Z \ N and z C \ D. The m = 1 case is of special interest: f [ 1] maps the exterior of the unit disk onto the inverted image of fd), which is a domain with bounded boundary. Actually, for fz) the interior.

19 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 19 whole-plane SLE κ map considered in Ref. [9] and here, f [ 1] z 1 ) is precisely the exterior whole-plane SLE κ map introduced in Ref. [3]. The moments, E f [m] ) z) p ) for m N \ {0}), as well as their associated integral means spectra were studied in Ref. [9]. Using Itô calculus, a PDE satisfied by these moments was derived for each value of m. The introduction of mixed p, q) moments allows us to circumvent these calculations in a unified approach. To see this, notice that As a consequence, f [m] ) z) = z m 1 f z m )fz m ) 1 m 1. z q f [m] ) z) p f [m] z) q = z q+pm 1) f z m ) p fz m q p p+ ) m so that we identically have E z q f ) [m] ) z) p 58) = Gz m ; p, q f [m] z) q m ), 59) with the notation, 60) q m = q m p, q) := p + q p m, Gz; p, q) := Gz, z) = E z q f ) z) p, fz) q where we have made explicit the dependence on the p, q) parameters of the SLE moduli one-point function 56) introduced in Section 3. From Theorem 4., we immediately get the following. Theorem 5.1. Let f [m] be the m-fold whole-plane SLE κ map, m Z \ {0}, with z D for m > 0 and z C \ D for m < 0. Then, E z q f ) [m] ) z) p = 1 zm ) α 1 z m ) α, f [m] z) q 1 z z) m ) κ α for p, q) belonging to the m-dependent parabola R [m], given in parametric form by p = + κ ) α κ α, q = m + + κ ) 61) α κ m + 1)α, α R. In Cartesian coordinates, an equivalent statement is with q = m + 1)p m + κ 4κ α = m + 1)p q m ), 1 + κ 4 + κ ± ) 4 + κ) 8κp, p, 4 + κ), 8κ

20 or, LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 0 p = q q m m + κ m κ ± ) m κ) m + 1) 4κ 8m + 1)κq, m κ) 8m + 1)κ. As for logarithmic coefficients, first observe that trivially, 6) log f [m] z) = 1 z m log fzm ) z. m From this, and Theorem 1.1, we thus get Corollary 5.. Let f [m] z) be the m-fold whole-plane SLE map and 63) log f [m] z) z then = n 1 γ [m] n z n ; { 1 E γ n [m] n = mk, k 1 ) = n 0 otherwise. We can also see this result as a corollary of Theorem 5.1, which, for the logarithmic case p = q, and for any value of m, yields p = q = for κ = as the only integrable case. 6. Integral means spectrum 6.1. Introduction. In this section we aim at generalizing to the setting of the present work the integral means spectrum analysis of Refs. [3] and [9] see also [18, 19, 0]) concerning the whole-plane SLE. The original work by Beliaev Smirnov [3] dealt with the exterior version, whereas Ref. [9] and this work concern the interior case. We thus look for the singular behavior of the integral, ) f z) p 64) E dz, fz) q r D for r 1, where f stands for the interior whole-plane SLE map at time zero). The integral means spectrum βp, q) corresponding to this generalized moment integral is the exponent such that ) f z) p 65) E dz r 1 ) 1 r) βp,q), fz) q r D in the sense of the equivalence of the logarithms of both terms. As mentioned in Section 5, it is interesting to remark that the map ˆf := f [ 1], ζ C \ D f [ 1] ζ) := 1/f1/ζ),

21 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 1 is just the exterior whole-plane map from C \ D to the slit plane considered by Beliaev and Smirnov in Ref. [3]. We identically have for 0 < r < 1: 66) E ˆf ) ) f ζ) p dζ = r p z) p E dz. fz) p r 1 D We thus see that the standard integral mean of order p, q = 0) for the exterior whole-plane map studied in Ref. [3] coincides up to an irrelevant power of r) with the p, q) integral mean 64) for q = p, for the interior whole-plane map. Remark 6.1. Exterior-Interior Duality. More generally, we obviously have 67) E ˆf ) ζ) p ) f dζ = r r 1 q D ˆfζ) p z) p E dz, r D fz) p q so that the p, q ) exterior integral means spectrum coincides with the p, q) interior integral means spectrum for q + q = p. In particular, the p, 0) interior derivative moments studied in Ref. [9] correspond to the p, p) mixed moments of the Beliaev Smirnov exterior map. Hence the general setting introduced in this work unifies the integral means spectrum studies of Refs. [3] and [9] in a broader framework, that also covers the p = q = q logarithmic case, as well as the integral means of the map f or ˆf) itself, in the 0, q) or 0, q)) case. 6.. Modified One-Point Function. Let us now consider the modified SLE moduli one-point function, 68) F z, z) := 1 ) f z) p Gz, z) = E. z q fz) q r D Because of Eq. 57), it obeys the modified PDE, 69) PD)[F z, z)] = κ z z ) F 1 + z 1 + z z F z F [ 1 z 1 ] z p + 1 z) p 1 z) + p q F z, z) = 0, which, of course, differs from Eq. 57). We can rewrite it as PD)[F z, z)] = κ z z ) F 1 + z 1 + z 70) z F z F [ 1 z 1 ] z 1 p 1 z) z) + σ 1 F = 0, in term of the important new parameter, 71) σ := q/p 1. This PDE then exactly coincides with Eq. 106) in Ref. [9], where σ was meant to represent ±1, whereas here σ R.

22 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE The value σ = +1 corresponds to the original Beliaev Smirnov case, where the integral means spectrum successively involves three functions [3]: 7) β tip p, κ) := p κ 4 + κ) 8κp), 73) 74) for p p 0κ) := 1 3κ 8 ; β 0 p, κ) := p κ 4κ 4 + κ 4 + κ) 8κp), for p 0κ) p p 0 κ); 75) 4 + κ) β lin p, κ) := p 16κ, 34 + κ) 76) for p p 0 κ) :=. 3κ As shown in Refs. [9, 18, 19, 0] in the σ = 1 interior case, because of the unboundedness of the interior whole-plane SLE map, there exists a phase transition at p = p κ), with p κ) := κ) 4 ) 4 + κ) 16κ + 4 = 1 ) 77) 4 + κ) κ) ). 3κ The integral means spectrum is afterwards given by 78) βp, κ) := 3p κp, for p p κ). Since p κ) < p 0 κ) 76), this transition precedes and supersedes the transition from the bulk spectrum 74) towards the linear behavior 75). The singularity analysis given in Ref. [9] led us to introduce the σ-dependent function 79) β σ +p, κ) = 1 σ)p σκp ). For σ = 1, it recovers the integral means spectrum 78) above for the interior whole-plane SLE, while for σ = +1 it introduces a new spectrum, 80) β +1) + p, κ) = p κp ), the relevance of which for the exterior whole-plane SLE case is analyzed in a joint work of two of us with D. Beliaev []. For general real values of σ 71), we can rewrite 79) as a function of p, q, κ), 81) β σ +p, κ) = β 1 p, q; κ) := 3p q κp q). We claim that the spectrum generated by the integral means 64) in the general p, q) case will involve the standard multifractal spectra 7), 74), 75), that are

23 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 3 independent of q, and also the new p, q)-dependent multifractal spectrum 81). Phase transitions between these spectra will occur along lines drawn in the real p, q) plane. The main reason for the occurence of 81), to be presented in a future work, is that the analysis performed in Ref. [9], Section 4, for determining the integral means spectrum, in particular the range of validity of 79) and the corresponding proofs, can be adapted for general values of the σ parameter. Here, we shall simply describe the corresponding partition of the p, q) plane into the respective domains of validity of the four spectra above. We thus need to determine the boundary curves where pairs possibly triplets) of these spectra coincide, which are signaling the onset of the respective transitions Phase transition lines. The best way is perhaps to recall the analytical derivation of the various multifractal spectra as done in Ref. [9], which was based on the use of functions A 38), B 39) and C 40). It will be convenient to use the notation [9], 8) such that for σ = q/p 1 71), A σ p, γ) := κ γ + γ σp, 83) A σ p, γ) = Ap, q; γ) = p q + γ κ γ, as well as 84) 85) 86) Bq, γ) = q 3 + κ ) γ + κγ, Cp, γ) = κ γ + + κ ) γ p, βp, γ) := κ γ Cp, γ) = κγ + κ ) γ + p, where the last function, βp, γ), is the so-called spectrum function [9]. Recall also that this function possesses an important duality property [9], 87) βp, γ) = βp, γ ), γ + γ := κ + 1. Remark 6.. The B S spectrum parameter γ 0, and bulk spectrum 74) β 0 := βp, γ 0 ), corresponding to Eqs. 11) and 1) in Ref. [3]) are obtained from the equations see Ref. [9]), 88) Cp, γ 0 ) = 0; β 0 = βp, γ 0 ) = κγ 0/. The two solutions to 88) are γ 0 ± p) as in Eq. 6), where the lower branch γ 0 := γ0 is the one selected for the bulk spectrum, β 0 p) = 1 κγ 0 p). This spectrum 74) is defined only to the left of a vertical line in the p, q) plane, as given by see Fig. 3) } 4 + κ) 89) 0 := {p =, q R. 8κ

24 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 4 Remark 6.3. The σ-dependent spectrum 79) is obtained from the equations 90) A σ p, γ) = 0; βp, γ) = κγ / Cp, γ). The solutions to Eq. 90) are 91) 9) γ σ ±p) = 1 κ 1 ± 1 σκp ), β σ ±p) = 1 σ)p κ γσ ±p) = 1 σ)p 1 1 ± 1 σκp ). The multifractal spectrum 79) is then given by the upper branch β σ +p) [9]. Note also that this spectrum is defined only for σκp 1, hence for points in the p, q) plane below the oblique line Fig. 3): 93) 1 := { p, q) R, q = p + 1/κ }. q D T D 1 P 0 T 1 T 0 p 0 ) p Figure 3. Red parabola R 95) and green parabola G 98) for κ = 6). From the intersection point P 0 100) originate the two half)-lines D 0 10) and D 1 103). The bulk spectrum β 0 p) and the generalized spectrum β 1 p, q) coincide along the arc 96) of red parabola between its tangency points T 0 and T 1 with 0 and 1 thick red line). They also coincide along the infinite left branch 99) of the green parabola, up to its tangency point T to 1 thick green line). The β 0 p) spectrum and the linear one β lin p) coincide along D 0, whereas β 1 p, q) and β lin p) coincide along D 1.

25 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE The Red Parabola. The parabola R of Theorems 3.1 and 4., which we shall hereafter call and draw in) red see Fig. 3), is given by the simultaneous conditions, 94) A σ p, γ) = Ap, q, γ) = 0, Cp, γ) = 0, hence also Bq, γ) = 0, which recovers the parametric form 4) p = p R γ) := + κ ) γ κ γ, 95) q = q R γ) := 3 + κ ) γ κγ, γ R. By construction, the associated spectrum βp, γ) is therefore both of the B S type, β 0 ± p), and of the novel type, β±p). σ We successively have: 96) γ = γ σ p) = γ 0 p); β σ p) = β 0 p), γ, 1/κ], γ = γ σ +p) = γ 0 p); β σ +p) = β 0 p), γ [1/κ, /κ + 1/], γ = γ σ +p) = γ + 0 p); β σ +p) = β + 0 p), γ [/κ + 1/, + ), where the change of analytic branch from the first to the second line corresponds to a tangency at T 1 of the red parabola to the boundary line 1, whereas the change from second to third corresponds to a tangency at T 0 to the vertical boundary line 0. The interval where the multifractal spectra coincide, i.e., when β+p) σ = β0 p), is thus given by line 96) in the equations above. In Cartesian coordinates, the red parabola R 95) has for equation 5) The Green Parabola. A second parabola in the p, q) plane, hereafter called green see Fig. 3) and denoted by G, is such that the multifractal spectra β 0 p) and β σ +p) = βp, q; κ) coincide on part of it. We use the duality property 87) of the spectrum function [9], and set the simultaneous seed conditions, 97) A σ p, γ ) = Ap, q, γ ) = 0, Cp, γ ) = 0, γ + γ = /κ + 1/, where γ and γ are dual of each other and such that βp, γ ) = βp, γ ). Eqs. 38) and 40) immediately give the parametric form for the green parabola, p = p G γ 4 + κ) ) := κ 8κ γ, 98) q = q G γ 4 + κ) ) := + γ κγ, γ R. 8κ Along this locus, we successively have: 99) γ = γ σ p), γ = γ + 0 p); β σ p) = β + 0 p), γ, 0], γ = γ σ p), γ = γ 0 p); β σ p) = β 0 p), γ [ 0, κ 1], γ = γ σ +p), γ = γ 0 p); β σ +p) = β 0 p), γ [ κ 1, + ),

26 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 6 where the changes of branches correspond to a tangency of the green parabola to 0 followed by a tangency to 1. The multifractal spectra coincide when β+p) σ = β0 p), which corresponds to the third line 99) in the equations above, i.e., to the domain where γ 1/κ Quadruple point. The intersection of the red and green parabolae 95) and 98) can be found by combining the seed equations 94) and 97). We find either γ = γ = 1/κ + 1/4, or γ = /κ + 1/4, γ = 1/4, which lead to the two intersection points, 34 + κ) 4 + κ)8 + κ) 100) P 0 : p 0 = p 0 κ) =, q 0 =, 3κ 16κ 8 + κ)8 + 3κ) 4 + κ)8 + κ) 101) P 1 : p 1 =, q 0 =. 3κ 16κ Note that these points have same ordinate, while the abscissa of the left-most one, P 0, is p 0 κ) 76), where the integral means spectrum transits from the B S bulk form 74) to its linear form 75). Through this intersection point P 0 further pass two important straight lines in the p, q) plane. Definition 6.4. D 0 and D 1 are, respectively, the vertical line and the slope one line passing through point P 0, of equations 10) 103) D 0 := {p, q) : p = p 0 }, D 1 := {p, q) : q p = q 0 p 0 = A key property of D 1 is the following. The difference, } 16 κ. 3κ 104) β 1 p, q; κ) β lin p, κ) = 1 κ κ κp q)), is always positive, and vanishes only on line D 1, where 105) p, q) D 1, β 1 p, q; κ) = β lin p, κ) = p 4 + κ) 16κ The Blue Quartic. A third locus, the blue quartic Q, will also play an important role, that is where the B S tip-spectrum, β tip p; κ) 7), coincides with the novel spectrum, β+p) σ = β 1 p, q; κ). The tip spectrum is given by β tip p; κ) = βp, γ 0 ) γ 0 1, where γ 0 is solution to Cp, γ 0 ) = 0 and such that the tip contribution is positive, γ [3, 9]; this corresponds to the tip condition 73) [3]. In the p, q) plane, this descibes the domain to the left of the straight line D 0 Fig. 4), defined by 106) D 0 := {p, q) : p = p 0κ) = 1 3κ/8}.

27 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 7 D 0 q /8 p Q 0 0 Figure 4. The blue quartic 113 for κ = 6. It intersects the green parabola at point Q 0 117) and the red parabola at point Q 1 116) not marked), both of abscissa p 0κ) = 1 3κ/8. The generalized spectrum is given by β σ +p) = βp, γ) where γ is solution to A σ p, γ) = 0. We therefore look for simultaneous solutions to the seed equations, 107) βp, γ) = βp, γ 0 ) γ 0 1, γ , A σ p, γ) = 0, Cp, γ 0 ) = 0. Using Eq. 83), we first find, as for the red and green parabolae, 108) q p = γ κ γ, and from 86) and 85), by substitution in the above, 109) p q + 1 = κ 4 γ + γ 0), 110) 4 + κ Solving for γ 0 in terms of γ gives 111) γ 0 = γ ± 0 γ κγ 1 = 8 + κ γ 0 κγ 0. := 8 + κ 4κ ± 1 κ 1 γ), 11) γ) := 4κ γ κ4 + κ)γ κ) + 4κ,

28 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 8 with γ) > 0, γ R. The tip relevance inequality in 107), γ , implies the choice of the negative branch in 111): γ 0 = γ0. We thus get the desired explicit parameterization of that branch of the quartic, 113) p = p Q γ) := κ κ 4 ) γ κ γ γ), q = q Q γ) := p Q γ) + γ κ γ, γ R. Remark 6.5. Note that because of the very choice to parameterize the parabolae and the quartic by γ, such that A 83) vanishes, Eq. 108) holds for each of the pairs of parametric equations. We successively have along the branch 113) of the blue quartic: 114) 115) γ = γ σ p); β σ p) = β tip p), γ, 1/κ], γ = γ σ +p); β σ +p) = β tip p) < β 0 p), γ [1/κ, 1 + /κ], γ = γ σ +p); β σ +p) = β tip p) β 0 p), γ [1 + /κ, + ), The intersection of the blue quartic 113) with the red parabola R 95) is located at 116) Q 1 : p 0 = 1 3κ 8, q = κ); γ = γ 0 = 1, followed by a second intersection at the origin, p = q = 0, for γ = κ and γ 0 = 0. The intersection of the blue quartic 113) with the green parabola G 98) is located at 117) Q 0 : p 0 = 1 3κ 8, q 0 := 7κ 8 ; γ = γ = 1 + κ, γ 0 = 1. Notice that these two intersection points have same abscissae, p 0κ) 73), where the transition for γ 0 = 1 from the B S bulk spectrum to the tip spectrum takes place. They are found by combining Eqs. 94) or Eqs. 97) with 107). The tip spectrum and the generalized one coincide in both γ-intervals 114) and 115), which together parameterize the branch of the quartic located below its contact with 1 see Fig. 4). Because of the tip relevance condition 73), only the interval 115) describing the lower infinite branch of the quartic located to the left of Q 0 will matter for the integral means spectrum Whole-plane SLE κ generalized spectrum Summary. Let us briefly summarize the results of Section 6.3. We know from Eq. 96) that the B S bulk spectrum β 0 p) and the mixed spectrum β 1 p, q) coincide along the finite sector of parabola R located between tangency points T 0 and T 1 Fig. 3). From Eq. 99), we also know that they coincide along the infinite left branch of parabola G below the tangency point T Fig. 3).

29 LOGARITHMIC COEFFICIENTS AND MULTIFRACTALITY OF WHOLE-PLANE SLE 9 The linear bulk spectrum β lin p) coincides with β 0 p) along line D 0 and supersedes the latter to the right of D 0 Fig. 3). We know from 105) that β lin p) and β 1 p, q) coincide along the line D 1 Fig. 3). The tip spectrum β tip p) coincides with β 0 p) along line D 0, and supersedes it to the left of D 0. We finally know from Eq. 115) that this tip spectrum β tip p) coincides with β 1 p, q) along the lower branch of the blue quartic located below point Q 0 117) Fig. 4). The only possible scenario which thus emerges to construct the average generalized integral means spectrum by a continuous matching of the 4 different spectra along the phase transition lines described above, is the partition of the p, q) plane in 4 different regions as indicated in Fig. 5: a part I) to the left of D 0 and located above the blue quartic up to point Q 0, where the average integral means spectrum is β tip p); an upper part II) bounded by lines D 0, D 0, and located above the section of the green parabola between points Q 0 and P 0, where the spectrum is given by β 0 p); an infinite wedge III) of apex P 0 located between the upper half-lines D 0 and D 1, where the spectrum is given by β lin p); a lower part IV) whose boundary is the blue quartic up to point Q 0, followed by the arc of green parabola between points Q 0 and P 0, followed by the half-line D 1 above P 0 where the spectrum is β 1 p, q). The two wings T 1 P 0 and P 0 T 0 of the red parabola Fig. 3), where we know from Theorem 4. that the average spectrum is given by β 0 p) = β 1 p, q), can thus be seen as the respective extensions of region IV into II and of region II into IV. This is summarized by the following proposition. Proposition 6.1. The separatrix curves for the generalized integral means spectrum of whole-plane SLE κ are in the p, q) plane Fig. 5): i) the vertical half-line D 0 above P 0 = p 0, q 0 ) 100), where p 0 = 34 + κ) /3κ, q 0 = 4 + κ)8 + κ)/16κ; ii) the unit slope half-line D 1 originating at P 0, whose equation is q p = 16 κ )/3κ with p p 0 ; iii) the section of green parabola, with parametric coordinates p G γ), q G γ) ) 98) for γ [1/4+1/κ, 1+/κ], between P 0 and Q 0 = p 0, q 0) 117), where p 0 = 1 3κ/8, q 0 = 7κ/8; iv) the vertical half-line D 0 above point Q 0 ; v) the branch of the blue quartic from Q 0 to, with parametric coordinates p Q γ), q Q γ) ) 113) for γ [1 + /κ, + ) The B S line. As mentioned above, the whole-plane SLE case studied by Beliaev and Smirnov corresponds to the q = p line. Because of Eq. 5), it intersects the red parabola R only at p = 0. The green parabola G 98) has for