Abstract
We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some integrability curves in the (p,q) moment plane, which depend continuously on the SLE parameter kappa. The generalization of this integrability property to the m-fold transform of f is also given. We define a generalized integral means spectrum corresponding to the
Results & Lemmas (33)
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Proposition 1.1.
Proposition 1.1. Let (ft(z))t≥0, z ∈D, be the interior Schramm–Loewner whole- plane process driven by λ(t) = ei√κBt in Eq. (2). We write…
Proposition 1.1. Let (ft(z))t≥0, z ∈D, be the interior Schramm–Loewner whole- plane process driven by λ(t) = ei√κBt in Eq. (2). We write (9) ft(z) = et z + X n≥2 an(t)zn . and for its logarithm, (10) log e−tft(z) z = 2 X n≥1
Theorem 1.2.
Theorem 1.2. Let f(z):= f0(z) be the time 0 unbounded whole-plane SLEκ map, in the same setting as in Proposition 1.1, such that log f(z) z…
Theorem 1.2. Let f(z) := f0(z) be the time 0 unbounded whole-plane SLEκ map, in the same setting as in Proposition 1.1, such that log f(z) z = 2 X n≥1 γnzn;
Theorem 1.3.
Theorem 1.3. Let f be the interior whole-plane SLEκ map, in the same setting as in Theorem 1.2; then for κ = 2, E zf ′(z) f(z)
Theorem 1.3. Let f be the interior whole-plane SLEκ map, in the same setting as in Theorem 1.2; then for κ = 2, E zf ′(z) f(z)
Theorem 1.3
Theorem 1.3 is actually a consequence of Theorems 3.1 and 4.2 of Sections 3 and 4 below, which give expressions in closed form for the…
Theorem 1.3 is actually a consequence of Theorems 3.1 and 4.2 of Sections 3 and 4 below, which give expressions in closed form for the mixed moments, (12) (a) E (f ′(z))p/2 (f(z))q/2 ; (b) E |f ′(z)|p |f(z)|q , along an integrability curve R, which is a parabola in the (p, q) plane depending
Theorem 1.7.
Theorem 1.7. Define the functions: βtip(p; κ):= −p −1 + 1 4(4 + κ − p (4 + κ)2 −8κp), β0(p; κ):= −p + 4 + κ 4κ (4 + κ − p (4 + κ)2 −8κp),…
Theorem 1.7. Define the functions: βtip(p; κ) := −p −1 + 1 4(4 + κ − p (4 + κ)2 −8κp), β0(p; κ) := −p + 4 + κ 4κ (4 + κ − p (4 + κ)2 −8κp), βlin(p; κ) := p −(4 + κ)2 16κ , β1(p, q; κ) :=3p −2q −1 2 −1 2
Proposition 2.1.
Proposition 2.1. Let f(z) = f0(z) be the interior whole-plane SLE2 map at time 0, in the same setting as in Proposition 1.1; we then have E…
Proposition 2.1. Let f(z) = f0(z) be the interior whole-plane SLE2 map at time 0, in the same setting as in Proposition 1.1; we then have E zf ′(z) f(z) = 1 −z. The method explained here will allow us to swiftly move to more complicated cases in the next sections. Let us then introduce (15) G(z) := E zf ′(z) f(z)
Lemma 2.3.
Lemma 2.3. The limit in law, limt→+∞et ˜ft(z), exists, and has the same law as the (time zero) interior whole-plane random map f0(z): lim…
Lemma 2.3. The limit in law, limt→+∞et ˜ft(z), exists, and has the same law as the (time zero) interior whole-plane random map f0(z): lim t→+∞et ˜ft(z) (law) = f0(z). Let us now turn to the proof of Proposition 2.1.
Theorem 3.1.
Theorem 3.1. Let f(z) = f0(z) be the interior whole-plane SLEκ map at time zero, in the same setting as in Proposition 1.1. Consider the…
Theorem 3.1. Let f(z) = f0(z) be the interior whole-plane SLEκ map at time zero, in the same setting as in Proposition 1.1. Consider the curve R, defined parametrically by p = −κ 2γ2 + 2 + κ 2 γ, 2p −q = 1 + κ 2
Lemma 4.1.
Lemma 4.1. The space of formal series F(z1, ¯z2) = P k,ℓ∈N ak,ℓzk 1 ¯zℓ 2, with complex coefficients and that are solutions of the PDE (56),…
Lemma 4.1. The space of formal series F(z1, ¯z2) = P k,ℓ∈N ak,ℓzk 1 ¯zℓ 2, with complex coefficients and that are solutions of the PDE (56), is one-dimensional.
Theorem 4.2.
Theorem 4.2. Let f(z) = f0(z) be the interior whole-plane SLEκ map in the setting of Proposition (1.1); then, for (p, q) belonging to the…
Theorem 4.2. Let f(z) = f0(z) be the interior whole-plane SLEκ map in the setting of Proposition (1.1); then, for (p, q) belonging to the parabola R defined in Theorem 3.1 by Eqs. (25) or (26) or (27), and for any pair (z1, z2) ∈D × D, E z q 2 1 (f ′(z1)) p 2 (f(z1)) q 2
Corollary 4.3.
Corollary 4.3. In the same setting as in Theorem 4.2, we have for z ∈D, E |z|q |f ′(z)|p |f(z)|q = (1 −z)γ(1 −¯z)γ (1 −z¯z)β, β = κ…
Corollary 4.3. In the same setting as in Theorem 4.2, we have for z ∈D, E |z|q |f ′(z)|p |f(z)|q = (1 −z)γ(1 −¯z)γ (1 −z¯z)β , β = κ 2γ2, for γ = γ± 0 (p) := 1 2κ
Corollary 4.4.
Corollary 4.4. The interior whole-plane SLEκ map has the integrable moments, E f(z1) z1 (2+κ)(4+κ) 4κ " f(z2) ¯z2 # (2+κ)(4+κ) 4κ …
Corollary 4.4. The interior whole-plane SLEκ map has the integrable moments, E f(z1) z1 (2+κ)(4+κ) 4κ " f(z2) ¯z2 # (2+κ)(4+κ) 4κ = (1 −z1)
Corollary 4.5.
Corollary 4.5. The interior whole-plane SLEκ map has the integrable logarith- mic derivative two-point function, E z1 f ′(z1) f(z1) …
Corollary 4.5. The interior whole-plane SLEκ map has the integrable logarith- mic derivative two-point function, E z1 f ′(z1) f(z1) 2+κ 2κ " ¯z2 f ′(z2) f(z2) # 2+κ 2κ = (1 −z1)
Theorem 1.3
Theorem 1.3 describes the κ = 2 case of the latter result. 4.4. Generalization to processes with m-fold symmetry. The moments, E(|(f…
Theorem 1.3 describes the κ = 2 case of the latter result. 4.4. Generalization to processes with m-fold symmetry. The moments, E(|(f [m])′(z)|p) (for m ∈N \ {0}), as well as their associated integral means spectra were studied in Ref. [14]. Using Itˆo 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 for m ∈Z \ {0}. To see this, notice that (f [m])′(z) = zm−1f ′(zm)f(
Theorem 4.6.
Theorem 4.6. Let f [m] be the m-fold whole-plane SLEκ 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…
Theorem 4.6. Let f [m] be the m-fold whole-plane SLEκ 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 |f [m](z)|q = (1 −zm)α(1 −¯zm)α (1 −(z¯z)m) κ 2 α2 ,
Corollary 4.7.
Corollary 4.7. Let f [m](z) be the m-fold whole-plane SLE2 map and (64) log f [m](z) z = 2 X n≥1 γ[m] n zn; then E(|γ[m] n |2) = 1 2n2
Corollary 4.7. Let f [m](z) be the m-fold whole-plane SLE2 map and (64) log f [m](z) z = 2 X n≥1 γ[m] n zn; then E(|γ[m] n |2) = 1 2n2
Theorem 4.2
Theorem 4.2 that the average spectrum is given by β0(p) = β1(p, q), can thus be seen as the respective extensions of region IV into II and…
Theorem 4.2 that the average spectrum is given by β0(p) = β1(p, q), can thus be seen as the respective extensions of region IV into II and of region II into IV. The validity of this geometrical analysis of the phase diagram of Figure 2, associated with the generalized integral means spectrum of whole-plane SLEκ, is established in Theorem 1.7. 5.3.2. The B–S line. As mentioned above, the whole-plane SLE case studied by Beliaev and Smirnov corresponds to the q = 2p line. Because of Eq. (26), it in
Lemma 5.5.
Lemma 5.5. The average generalized integral means spectrum β(p, q) of whole- plane SLE is bounded below as β(p, q) ≥β1(p, q) in E−∪I,…
Lemma 5.5. The average generalized integral means spectrum β(p, q) of whole- plane SLE is bounded below as β(p, q) ≥β1(p, q) in E−∪I, whereas β(p, q) ≤ β1(p, q) in E+.
Proposition 5.6.
Proposition 5.6. Consider in the (p, q) plane the upward wedge-like domain delimited by lines D0 and D1 interecting at P0 (Figs. 2 and 6).…
Proposition 5.6. Consider in the (p, q) plane the upward wedge-like domain delimited by lines D0 and D1 interecting at P0 (Figs. 2 and 6). In this domain, the average generalized integral means spectrum has the linear form βlin (72).
Proposition 5.7.
Proposition 5.7. Consider the infinite domain of the (p, q) plane above the infinite upper branch (99) of parabola G (96) located below point…
Proposition 5.7. Consider the infinite domain of the (p, q) plane above the infinite upper branch (99) of parabola G (96) located below point P0, and to the left of the half-line D0 above P0 (Fig. 6). The average generalized integral means spectrum β(p, q) is given in this domain by the standard bulk spectrum β0(p) for p ≥−1 −3κ/8 or by the tip version βtip(p) in the opposite case.
Proposition 4.1
Proposition 4.1 there. Two important quantities are defined as a(p):=γ0(p) −γσ +(p) = γ0(p) −1 κ −1 κ p 1 −2σκp, (122) b(p):=γ0(p) −γσ −(p)…
Proposition 4.1 there. Two important quantities are defined as a(p) :=γ0(p) −γσ +(p) = γ0(p) −1 κ −1 κ p 1 −2σκp, (122) b(p) :=γ0(p) −γσ −(p) = γ0(p) −1 κ + 1 κ p 1 −2σκp, (123)
Proposition 5.9.
Proposition 5.9. In domain D, the average generalized integral means spectrum is given by β(p, q) = max βtip(p), β1(p, q).
Proposition 5.9. In domain D, the average generalized integral means spectrum is given by β(p, q) = max{βtip(p), β1(p, q)}.
Corollary 5.10.
Corollary 5.10. Since in D the blue quartic Q below Q0 is the separatrix for βtip(p) = β1(p, q), the average generalized integral means…
Corollary 5.10. Since in D the blue quartic Q below Q0 is the separatrix for βtip(p) = β1(p, q), the average generalized integral means spectrum is given by βtip(p) in the infinite thin wedge W in D of apex Q0, located to the left of line D′ 0, inbetween the green parabola G and the blue quartic Q, and by β1(p, q) in the remaining part D \ W of D (Fig. 6).
Lemma 5.11.
Lemma 5.11. For (p, q) ∈D, there is r0 < 1 such that ψ = ςψ0 + ψ1 > 0 for all z such that r0 < |z| < 1.
Lemma 5.11. For (p, q) ∈D, there is r0 < 1 such that ψ = ςψ0 + ψ1 > 0 for all z such that r0 < |z| < 1.
Lemma 5.13.
Lemma 5.13. The set of equations, b = n+ 1 2, n ∈N, where g0(0) = 0, is realized on the set of parabola branches P+ n as defined in Def.…
Lemma 5.13. The set of equations, b = n+ 1 2, n ∈N, where g0(0) = 0, is realized on the set of parabola branches P+ n as defined in Def. 5.12. Inside domain D, this yields the set Tκ := ∪n∈JκP+ n ∩D, where Jκ := {n ∈N, 0 ≤n ≤⌊2κ−1⌋}.
Lemma 5.17.
Lemma 5.17. For (p, q) ∈D and (p, q) /∈Tκ, there is r0 < 1 such that P(D)[ψℓδ] for ψ (128) has a constant sign in the annulus r0 < |z| < 1,…
Lemma 5.17. For (p, q) ∈D and (p, q) /∈Tκ, there is r0 < 1 such that P(D)[ψℓδ] for ψ (128) has a constant sign in the annulus r0 < |z| < 1, which depends only on that of δ.
Lemma 4.2
Lemma 4.2] makes a crucial use of the inequality β1 < βtip. Here, this is precisely valid in the wedge W ⊂D (recall Corollary 5.10), which…
Lemma 4.2] makes a crucial use of the inequality β1 < βtip. Here, this is precisely valid in the wedge W ⊂D (recall Corollary 5.10), which thus establishes Lemma 5.17 in W. In D \ W, we need a different, more general argument.
Proposition 5.18.
Proposition 5.18. Define ˜D as the infinite domain located between the left branch (99) of the green parabola G stemming from point P0 and…
Proposition 5.18. Define ˜D as the infinite domain located between the left branch (99) of the green parabola G stemming from point P0 and the right branch of the red parabola R (91) starting from P0. Define ˆD ⊂˜D as the racket-shaped finite domain located above the straight line D3 of equation p −q = ˆp(κ) = 1 + κ 2,
Theorem 1.7
Theorem 1.7 and Fig. 2, in the following maximal domain of the (p, q)-plane. It is in Figure 6 the infinite domain above the frontier line…
Theorem 1.7 and Fig. 2, in the following maximal domain of the (p, q)-plane. It is in Figure 6 the infinite domain above the frontier line made by the union of the lower branch of the green parabola up to its intersection point with D3, obtained for γ′ = −1 in Eq. (97) as (4 −κ)(4 + 3κ)/8κ, (16 −7κ2)/8κ , of the segment of D3 from that point up to P3, of the branch of the red parabola R between P3 and P0, and finally of D1 from P0 up to infinity. This concludes the proof of
Theorem 1.7.
Theorem 1.7. □ 5.5. m-fold spectrum. For m ≥1, the generalized integral means spectrum β[m](p, q; κ), associated with the m-fold transform…
Theorem 1.7. □ 5.5. m-fold spectrum. For m ≥1, the generalized integral means spectrum β[m](p, q; κ), associated with the m-fold transform f [m] of the SLE whole-plane map, can be directly derived from the analysis given in Section 4.4. Definition 1.4 and identities (59) and (60) immediately imply that β[m](p, q; κ) = β[1](p, qm; κ), qm = qm(p, q) = (1 −1/m) p + q/m, (141) where β[1](p, q; κ) := β(p, q; κ) is the m = 1 average generalized integral means spectrum of whole-plane SLEκ studied above.
Theorem 5.20.
Theorem 5.20. (Figure 9) Separatrix curves for the generalized integral means spectrum β[m](p, q; κ) of the m-fold whole-plane SLEκ are…
Theorem 5.20. (Figure 9) Separatrix curves for the generalized integral means spectrum β[m](p, q; κ) of the m-fold whole-plane SLEκ are given, for m ≥1, by the same as in Theorem 1.7 for m = 1, provided that one replaces there, • D0 by D[m] 0 , P0 by P [m] 0 = (p0, q[m] 0 ), q0 by q[m] 0 := p0 + m(16 −κ2)/32κ;
Theorem 6.1.
Theorem 6.1. Let f be holomorphic and injective in the unit disk. For p ∈ R+, q ∈R such that q < min 2, 5 4p −1 2, there exists a constant…
Theorem 6.1. Let f be holomorphic and injective in the unit disk. For p ∈ R+, q ∈R such that q < min{2, 5 4p −1 2}, there exists a constant C > 0 such that (147) Z 2π 0 |f ′(reiθ)|p |f(reiθ)|q dθ ≤ C (1 −r)3p−2q−1. The universal spectrum is therefore finite and such that B(p, q) ≤3p −2q −1, at least in the domain D0 := {0 ≤p, q < min{2, 5 4p−1 2}} of Theorem 6.1. In that
Theorem 6.2.
Theorem 6.2. The universal generalized spectrum is given by B(p, q) = max B(p), 3p −2q −1, where B(p) is the universal spectrum for bounded…
Theorem 6.2. The universal generalized spectrum is given by B(p, q) = max{B(p), 3p −2q −1}, where B(p) is the universal spectrum for bounded univalent functions. This confirms the conclusions drawn above for the universal generalized spec- trum, the unknown remaining the position of p† and the form of B0(p) in the standard universal spectrum. References [1] K. Astala, B. Duplantier, and M. Zinsmeister, 2015. Unpublished manuscript. [2] D. Beliaev, B. Duplantier, and M. Zinsmeister. Integral Means
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