Imaginary Geometry and the Gaussian Free Field
|
|
|
- Roxanne Long
- 5 years ago
- Views:
Transcription
1 Imaginary Geometry and the Gaussian Free Field Jason Miller and Scott Sheffield Massachusetts Institute of Technology May 23, 2013 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
2 Part I: SLE Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
3 SLE κ Random, fractal curve in D C simply connected Introduced by Oded Schramm as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. (Percolation simulations due to Sam Watson) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
4 SLE κ Random, fractal curve in D C simply connected Introduced by Oded Schramm as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. (Percolation simulations due to Sam Watson) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
5 SLE κ Random, fractal curve in D C simply connected Introduced by Oded Schramm as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. (Percolation simulations due to Sam Watson) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
6 SLE κ η(s) η(t) g t g t (η(s)) Loewner s equation: if η is a non self-crossing path in H with η(0) R and g t is the Riemann map from the unbounded component of H \ η([0, t]) to H normalized by g t(z) = z + o(1) as z, then tg t(z) = 2 g t(z) W t where g 0(z) = z and W t = g t(η(t)). ( ) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
7 SLE κ η(s) η(t) g t g t (η(s)) Loewner s equation: if η is a non self-crossing path in H with η(0) R and g t is the Riemann map from the unbounded component of H \ η([0, t]) to H normalized by g t(z) = z + o(1) as z, then tg t(z) = 2 g t(z) W t where g 0(z) = z and W t = g t(η(t)). ( ) SLE κ in H: The random curve associated with ( ) with W t = κb t, B a standard Brownian motion. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
8 SLE κ η(s) η(t) g t g t (η(s)) Loewner s equation: if η is a non self-crossing path in H with η(0) R and g t is the Riemann map from the unbounded component of H \ η([0, t]) to H normalized by g t(z) = z + o(1) as z, then tg t(z) = 2 g t(z) W t where g 0(z) = z and W t = g t(η(t)). ( ) SLE κ in H: The random curve associated with ( ) with W t = κb t, B a standard Brownian motion. Other domains: apply conformal mapping. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
9 SLE κ SLE κ is a continuous path (κ 8, Rohde-Schramm κ = 8, Lawler, Schramm, Werner), Dimension: 1 + κ 8 (Beffara), Simple if κ (0, 4], self-touching if κ > 4, space-filling if κ 8. (Simulations due to Tom Kennedy) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
10 Sample path properties of the SLE trace Continuity; Hölder exponent Dimensions Multi-fractal spectra Natural parameterization Duality (path decompositions) CLE (loop version of SLE) SLE on multiply connected domains Reversibility Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
11 Reversibility of SLE κ An SLE κ process η from x to y. Question: If we run η in the reverse direction (i.e., from y to x), is it still an SLE κ? Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
12 Reversibility of SLE κ An SLE κ process η from x to y. Question: If we run η in the reverse direction (i.e., from y to x), is it still an SLE κ? Answer: Yes for κ 8. x y Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
13 A related question An SLE κ process η from x to y. Question: If we observe the range of η, can we determine the path? Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
14 A related question An SLE κ process η from x to y. Question: If we observe the range of η, can we determine the path? Answer: No for κ > 4. x y Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
15 Part II: SLE and the Gaussian Free Field Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
16 Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
17 Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Vector field e ih(x,y) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
18 Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Vector field e ih(x,y) A ray of h is a flow line of e ih, i.e. a solution to d η(t) = eih(η(t)) dt Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
19 Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Vector field e ih(x,y) A θ-angle ray of h is a flow line of e i(h+θ), i.e. a solution to d η(t) = ei(h(η(t))+θ) dt Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
20 Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Vector field e ih(x,y) A θ-angle ray of h is a flow line of e i(h+θ), i.e. a solution to d η(t) = ei(h(η(t))+θ) dt The rays of h vary smoothly and monotonically with θ and are non-intersecting Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
![Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Vector field e ih(x,y) A θ-angle ray of h is a flow line of e i(h+θ), i.e. a solution to d η(t) = ei(h(η(t))+θ) dt The rays of h vary smoothly and monotonically with θ and are non-intersecting.](/docs-images/90/102619992/images/21-0.jpg "Plan: Describe the interaction of the rays of the GFF Explain how this can be used to study SLE 3 2 1 0 1 2 3 3 2 1 0 1 2 3 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian")
21 Rays of a Smooth Function h smooth [h(x, y) = x 2 + y 2 ] Vector field e ih(x,y) A θ-angle ray of h is a flow line of e i(h+θ), i.e. a solution to d η(t) = ei(h(η(t))+θ) dt The rays of h vary smoothly and monotonically with θ and are non-intersecting. Plan: Describe the interaction of the rays of the GFF Explain how this can be used to study SLE Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
22 The Gaussian Free Field I The discrete Gaussian free field (DGFF) is a Gaussian random surface model. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
23 The Gaussian Free Field I The discrete Gaussian free field (DGFF) is a Gaussian random surface model. I Gaussian measure on functions h : D R for D Z2 and h D = ψ where I I Covariance: Green s function for discrete Mean Height: discrete harmonic ext. of ψ Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
24 The Gaussian Free Field I The discrete Gaussian free field (DGFF) is a Gaussian random surface model. I Gaussian measure on functions h : D R for D Z2 and h D = ψ where I I I Covariance: Green s function for discrete Mean Height: discrete harmonic ext. of ψ Density with respect to Lebesgue measure on R D :! 1 1X 2 exp (h(x) h(y )) Z 2 x y Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
25 The Gaussian Free Field I The discrete Gaussian free field (DGFF) is a Gaussian random surface model. I Gaussian measure on functions h : D R for D Z2 and h D = ψ where I I Covariance: Green s function for discrete Mean Height: discrete harmonic ext. of ψ I Density with respect to Lebesgue measure on R D :! 1 1X 2 exp (h(x) h(y )) Z 2 x y I Fine mesh limit: converges to the continuum GFF, i.e. the standard Gaussian wrt the Dirichlet inner product Z 1 (f, g ) = f (x) g (x)dx. 2π Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
26 Properties of the GFF Conformal invariance D, D C domains, ϕ: D D conformal transformation. If h is a GFF on D then h ϕ 1 is a GFF on D. ϕ D D Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
27 Properties of the GFF Conformal invariance D, D C domains, ϕ: D D conformal transformation. If h is a GFF on D then h ϕ 1 is a GFF on D. ϕ D D Markov property D C domain, h a GFF on D. D Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
28 Properties of the GFF Conformal invariance D, D C domains, ϕ: D D conformal transformation. If h is a GFF on D then h ϕ 1 is a GFF on D. ϕ D D Markov property D C domain, h a GFF on D. Fix S D. Conditional law of h D\S given h S is a GFF on D \ S plus the harmonic extension of h (D\S) to D \ S. S D Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
29 Constructing the rays of the GFF Suppose h is smooth on H and η is a ray of h with η(0) = 0, i.e. d η(t) = eih(η(t)) dt η Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
30 Constructing the rays of the GFF Suppose h is smooth on H and η is a ray of h with η(0) = 0, i.e. η determines the values of h along η and η is independent of the values of h off of η d η(t) = eih(η(t)) dt η Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
31 Constructing the rays of the GFF Sample a (simple) SLE κ process η in H from 0 to (i.e. κ (0, 4)) η Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
32 Constructing the rays of the GFF Sample a (simple) SLE κ process η in H from 0 to (i.e. κ (0, 4)) Given η, let h be a GFF on H \ η with boundary data on η given by what it would be if η were a flow line of e ih/χ where χ = 2 κ κ 2 η Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
33 Constructing the rays of the GFF Sample a (simple) SLE κ process η in H from 0 to (i.e. κ (0, 4)) Given η, let h be a GFF on H \ η with boundary data on η given by what it would be if η were a flow line of e ih/χ where χ = 2 κ κ 2 Theorem (Sheffield) The marginal law of h is that of a GFF on all of H η Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
34 Constructing the rays of the GFF Sample a (simple) SLE κ process η in H from 0 to (i.e. κ (0, 4)) Given η, let h be a GFF on H \ η with boundary data on η given by what it would be if η were a flow line of e ih/χ where χ = 2 κ κ 2 Theorem (Sheffield) The marginal law of h is that of a GFF on all of H Theorem (Dubédat) η is almost surely determined by h η Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
35 Multiple rays of the GFF A random closed set S coupled with a GFF h on a domain D is said to be local for h if the conditional law of h D\S given S is that of the sum of a zero-boundary GFF on D \ S and an independent harmonic function. S D Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
36 Multiple rays of the GFF A random closed set S coupled with a GFF h on a domain D is said to be local for h if the conditional law of h D\S given S is that of the sum of a zero-boundary GFF on D \ S and an independent harmonic function. S D Examples Deterministic closed sets. An SLE path coupled with h as a flow line drawn up to a stopping time. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
37 Multiple rays of the GFF A random closed set S coupled with a GFF h on a domain D is said to be local for h if the conditional law of h D\S given S is that of the sum of a zero-boundary GFF on D \ S and an independent harmonic function. A 1 A 2 D Examples Deterministic closed sets. An SLE path coupled with h as a flow line drawn up to a stopping time. Fact (Schramm, Sheffield) If A 1, A 2 are local sets for a GFF h which are conditionally independent given h, then A 1 A 2 is a local set for h. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
38 Multiple rays of the GFF A random closed set S coupled with a GFF h on a domain D is said to be local for h if the conditional law of h D\S given S is that of the sum of a zero-boundary GFF on D \ S and an independent harmonic function. A 1 A 2 D Examples Deterministic closed sets. An SLE path coupled with h as a flow line drawn up to a stopping time. Fact (Schramm, Sheffield) If A 1, A 2 are local sets for a GFF h which are conditionally independent given h, then A 1 A 2 is a local set for h. Moreover, the boundary behavior for the conditional law of h given A 1 A 2 agrees with that of h given A 1 near points points z A 1 with dist(z, A 2) > 0 and vice-versa. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
39 Multiple rays of the GFF A random closed set S coupled with a GFF h on a domain D is said to be local for h if the conditional law of h D\S given S is that of the sum of a zero-boundary GFF on D \ S and an independent harmonic function. A 1 A 2 D Examples Deterministic closed sets. An SLE path coupled with h as a flow line drawn up to a stopping time. Fact (Schramm, Sheffield) If A 1, A 2 are local sets for a GFF h which are conditionally independent given h, then A 1 A 2 is a local set for h. Moreover, the boundary behavior for the conditional law of h given A 1 A 2 agrees with that of h given A 1 near points points z A 1 with dist(z, A 2) > 0 and vice-versa. Leads to couplings with many SLE paths. Intersection points of paths cause trouble. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
40 Rays of e ih/χ, h GFF, χ [κ = 1/256] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
41 Rays of e ih/χ, h GFF, χ [κ = 1/32] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
42 Rays of e ih/χ, h GFF, χ 7.88 [κ = 1/16] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
43 Rays of e ih/χ, h GFF, χ = 3.75 [κ = 1/4] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
44 Rays of e ih/χ, h GFF, χ 2.47 [κ = 1/2] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
45 Rays of e ih/χ, h GFF, χ = 1.5 [κ = 1] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
46 Rays of e ih/χ, h GFF, χ 1.02 [κ = 3/2] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
47 Rays of e ih/χ, h GFF, χ = 0.71 [κ = 2] Angle π 2 Angle π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
48 Rays of e ih/χ starting from an interior point Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
49 Rays of e i h/χ, h = h + β log, h GFF, β < 0 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
50 Rays of e ih/χ, h GFF, χ 2.47 [κ = 1/2] Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
51 Rays of e ih/χ, h GFF, χ 2.47 [κ = 1/2] Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
52 Rays of e ih/χ, h GFF, χ 2.47 [κ = 1/2] Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
53 Rays of e ih/χ, h GFF, χ 2.47 [κ = 1/2] Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
54 Background Existence and uniqueness of couplings (η, h) of a GFF h and η SLE κ are studied in the works of Sheffield, Schramm-Sheffield, Dubédat, and Izyurov-Kytöla New Contributions: Developed a robust theory of flow line interaction to make the phenomena observed in the simulations precise General forms of SLE duality the SLE light cone SLE κ(ρ) processes are continuous (even when they hit the boundary) Important variant of SLEκ The drift for the driving function includes a linear combination of the Loewner evolution of a collection force points Double and cut point dimension of SLE Reversibility results Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
55 Flow line interaction θ 1>θ 2 η θ1 η θ2 x 1 x 2 h GFF on H, x 1, x 2 H with x 1 x 2. η θ1, η θ2 flow line of h with angles θ 1, θ 2 starting from x 1, x 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
56 Flow line interaction θ 1>θ 2 η θ1 η θ2 x 1 x 2 h GFF on H, x 1, x 2 H with x 1 x 2. η θ1, η θ2 flow line of h with angles θ 1, θ 2 starting from x 1, x 2 Theorem (M., Sheffield) If θ1 > θ 2, then η θ1 stays to the left of η θ2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
57 Flow line interaction θ 1>θ 2 θ 1=θ 2 η θ1 η θ2 η θ1 η θ2 x 1 x 2 x 1 x 2 h GFF on H, x 1, x 2 H with x 1 x 2. η θ1, η θ2 flow line of h with angles θ 1, θ 2 starting from x 1, x 2 Theorem (M., Sheffield) If θ1 > θ 2, then η θ1 stays to the left of η θ2 If θ1 = θ 2, then η θ1 merges with η θ2 upon intersecting Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
58 Flow line interaction θ 1>θ 2 θ 1=θ 2 θ 1<θ 2 η θ1 η θ2 η θ1 η θ2 η θ1 η θ2 x 1 x 2 x 1 x 2 x 1 x 2 h GFF on H, x 1, x 2 H with x 1 x 2. η θ1, η θ2 flow line of h with angles θ 1, θ 2 starting from x 1, x 2 Theorem (M., Sheffield) If θ1 > θ 2, then η θ1 stays to the left of η θ2 If θ1 = θ 2, then η θ1 merges with η θ2 upon intersecting If θ1 < θ 2, then η θ1 crosses η θ2 upon intersecting Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
59 Flow line interaction θ 1>θ 2 θ 1=θ 2 θ 1<θ 2 η θ1 η θ2 η θ1 η θ2 η θ1 η θ2 x 1 x 2 x 1 x 2 x 1 x 2 h GFF on H, x 1, x 2 H with x 1 x 2. η θ1, η θ2 flow line of h with angles θ 1, θ 2 starting from x 1, x 2 Theorem (M., Sheffield) If θ1 > θ 2, then η θ1 stays to the left of η θ2 If θ1 = θ 2, then η θ1 merges with η θ2 upon intersecting If θ1 < θ 2, then η θ1 crosses η θ2 upon intersecting In each case, the conditional law of η θ1 given η θ2 is an SLE κ(ρ) process Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
60 SLE κ (ρ) η(s) η(t) g t g t (η(s)) 0 W t Loewner ODE tg t(z) = 2 g t(z) W t where g 0(z) = z. SLE κ: W t = κb t where B is a standard Brownian motion Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
61 SLE κ (ρ) η(s) η(t) g t g t (η(s)) 0 v ρ W t V t ρ Loewner ODE tg t(z) = 2 g t(z) W t where g 0(z) = z. SLE κ: W t = κb t where B is a standard Brownian motion SLE κ(ρ): Force point v > 0 and a weight ρ R. dw t = κdb t + ρ 2 dt where dv t = dt, V 0 = v. W t V t V t W t Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
62 Continuity SLE κ (ρ) SLE κ is continuous (κ 8: Rohde-Schramm, κ = 8: Lawler-Schramm-Werner) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
63 Continuity SLE κ (ρ) SLE κ is continuous (κ 8: Rohde-Schramm, κ = 8: Lawler-Schramm-Werner) Theorem (M., Sheffield) SLE κ(ρ) processes are continuous. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
64 Continuity SLE κ (ρ) SLE κ is continuous (κ 8: Rohde-Schramm, κ = 8: Lawler-Schramm-Werner) Theorem (M., Sheffield) SLE κ(ρ) processes are continuous. Proof: (for SLE κ(ρ 1; ρ 2) for κ (0, 4)) Let η θ be the flow line with angle θ and fix θ 1 > 0 > θ 2. Fact: η θ1, η θ2, η 0 are continuous if they do not hit the boundary. Reason: Their law is mutually absolutely continuous with respect to SLE κ. The law of η 0 given η θ1 and η θ2 is an SLE κ(ρ 1; ρ 2) process. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
65 SLE Duality Fix κ (0, 4). The outer boundary of an SLE 16/κ process is described by a certain SLE κ process. Predicted by Duplantier Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
66 SLE Duality Fix κ (0, 4). The outer boundary of an SLE 16/κ process is described by a certain SLE κ process. Predicted by Duplantier Natural for certain values of κ, i.e. κ = 2 (LERW) and 16/κ = 8 (UST) Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
67 SLE Duality Fix κ (0, 4). The outer boundary of an SLE 16/κ process is described by a certain SLE κ process. Predicted by Duplantier Natural for certain values of κ, i.e. κ = 2 (LERW) and 16/κ = 8 (UST) Proved in various forms by Zhan and Dubédat Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
68 Duality in the GFF: the SLE Light Cone Flow lines with fixed angle π 2 and π 2. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
69 Duality in the GFF: the SLE Light Cone Flow lines with angle π 2 and π ; one direction change. 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
70 Duality in the GFF: the SLE Light Cone Flow lines with angle π 2 and π ; two direction changes. 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
71 Duality in the GFF: the SLE Light Cone Flow lines with angle π 2 and π ; three direction changes. 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
72 Duality in the GFF: the SLE Light Cone Flow lines with angle π 2 and π ; four direction changes. 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
73 Duality in the GFF: the SLE Light Cone Theorem (M., Sheffield): The set of all points accessible by SLE κ flow lines (κ (0, 4)) with angles restricted in [ π 2, π 2 ] is an SLE 16/κ process. Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
74 SLE 128 Light Cone Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
75 Dimensions Flow lines with angle π 2 and π 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
76 Dimensions Flow lines with angle π 2 and π 2 SLE κ for κ = 16/κ Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
77 Dimensions Theorem (M., Wu) The cut point dimension of SLE κ for κ (4, 8) is 3 3κ 8 Flow lines with angle π 2 and π 2 SLE κ for κ = 16/κ Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
78 Dimensions Theorem (M., Wu) The cut point dimension of SLE κ for κ (4, 8) is 3 3κ 8 Corollary of a result which gives the dimension of the intersection of flow lines with different angles. Flow lines with angle π 2 and π 2 SLE κ for κ = 16/κ Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
79 Dimensions Theorem (M., Wu) The cut point dimension of SLE κ for κ (4, 8) is 3 3κ 8 Corollary of a result which gives the dimension of the intersection of flow lines with different angles. Also gives the double point dimension is 2 (12 κ )(4 + κ )/(8κ ). Flow lines with angle π 2 and π 2 SLE κ for κ = 16/κ Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
80 Reversibility of SLE κ for κ 8 The time-reversal of an SLE κ is not an SLE κ for κ > 8 (Rohde-Schramm). Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
81 Reversibility of SLE κ for κ 8 Theorem (M., Sheffield) SLE κ( κ 2; κ 2) is reversible for κ Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
82 Reversibility of SLE κ for κ 8 Theorem (M., Sheffield) SLE κ( κ 2; κ 2) is reversible for κ 8. The time-reversal 4 4 of ordinary SLE κ is an SLE κ( κ 4; κ 4). 2 2 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
83 The SLE κ fan is reversible Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
84 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian Free Field May 23, / 30
An Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute. March 14, 2008
An Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute March 14, 2008 Outline Lattice models whose time evolution is not Markovian. Conformal invariance of their scaling
The dimer model: universality and conformal invariance. Nathanaël Berestycki University of Cambridge. Colloque des sciences mathématiques du Québec
The dimer model: universality and conformal invariance Nathanaël Berestycki University of Cambridge Colloque des sciences mathématiques du Québec The dimer model Definition G = bipartite finite graph,
Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM) and percolation (c) Loop-erased random walk (LERW) (d
Percolation, Brownian Motion and SLE Oded Schramm The Weizmann Institute of Science and Microsoft Research Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM)
Plan 1. Brownian motion 2. Loop-erased random walk 3. SLE 4. Percolation 5. Uniform spanning trees (UST) 6. UST Peano curve 7. Self-avoiding walk 1
Conformally invariant scaling limits: Brownian motion, percolation, and loop-erased random walk Oded Schramm Microsoft Research Weizmann Institute of Science (on leave) Plan 1. Brownian motion 2. Loop-erased
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees arxiv:130.4738v [math.pr] 6 Apr 017 Jason Miller and Scott Sheffield Abstract We establish existence and uniqueness
GEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE)
GEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE) Triennial Ahlfors-Bers Colloquium Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago 5734 S.
Fluctuations for the Ginzburg-Landau Model and Universality for SLE(4)
Fluctuations for the Ginzburg-Landau Model and Universality for SLE(4) Jason Miller Department of Mathematics, Stanford September 7, 2010 Jason Miller (Stanford Math) Fluctuations and Contours of the GL
Level lines of the Gaussian Free Field with general boundary data
Level lines of the Gaussian Free Field with general boundary data UK Easter Probability Meeting 2016 April 7th, 2016 The Gaussian Free Field GFF on domain D R 2 Gaussian random function. The Gaussian Free
Towards conformal invariance of 2-dim lattice models
Towards conformal invariance of 2-dim lattice models Stanislav Smirnov Université de Genève September 4, 2006 2-dim lattice models of natural phenomena: Ising, percolation, self-avoiding polymers,... Realistic
Fractal Properties of the Schramm-Loewner Evolution (SLE)
Fractal Properties of the Schramm-Loewner Evolution (SLE) Gregory F. Lawler Department of Mathematics University of Chicago 5734 S. University Ave. Chicago, IL 60637 lawler@math.uchicago.edu December 12,
QLE. Jason Miller and Scott Sheffield. August 1, 2013 MIT. Jason Miller and Scott Sheffield (MIT) QLE August 1, / 37
QLE Jason Miller and Scott Sheffield MIT August 1, 2013 Jason Miller and Scott Sheffield (MIT) QLE August 1, 2013 1 / 37 Surfaces, curves, metric balls: how are they related? FPP: first passage percolation.
Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve
Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve Hiroyuki Suzuki Chuo University International Workshop on Conformal Dynamics and Loewner Theory 2014/11/23 1 / 27 Introduction(1)
Advanced Topics in Probability
Advanced Topics in Probability Conformal Methods in 2D Statistical Mechanics Pierre Nolin Different lattices discrete models on lattices, in two dimensions: square lattice Z 2 : simplest one triangular
arxiv: v4 [math.pr] 1 Sep 2017
![arxiv: v4 [math.pr] 1 Sep 2017](/thumbs/76/73131309.jpg "arxiv: v4 [math.pr] 1 Sep 2017")
CLE PERCOLATIONS JASON MILLER, SCOTT SHEFFIELD, AND WENDELIN WERNER arxiv:1602.03884v4 [math.pr] 1 Sep 2017 Abstract. Conformal loop ensembles are random collections of loops in a simply connected domain,
2D Critical Systems, Fractals and SLE
2D Critical Systems, Fractals and SLE Meik Hellmund Leipzig University, Institute of Mathematics Statistical models, clusters, loops Fractal dimensions Stochastic/Schramm Loewner evolution (SLE) Outlook
Coupling the Gaussian Free Fields with Free and with Zero Boundary Conditions via Common Level Lines
Commun. Math. Phys. 361, 53 80 (2018) Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-018-3159-z Communications in Mathematical Physics Coupling the Gaussian Free Fields with Free and with
arxiv:math/ v1 [math.pr] 21 Dec 2001
![arxiv:math/ v1 [math.pr] 21 Dec 2001](/thumbs/92/109843926.jpg "arxiv:math/ v1 [math.pr] 21 Dec 2001")
Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk arxiv:math/0112246v1 [math.pr] 21 Dec 2001 Tom Kennedy Departments of Mathematics and Physics University of Arizona, Tucson, AZ, 85721
arxiv:math-ph/ v2 6 Jun 2005
On Conformal Field Theory of SLE(κ, ρ) arxiv:math-ph/0504057v 6 Jun 005 Kalle Kytölä kalle.kytola@helsinki.fi Department of Mathematics, P.O. Box 68 FIN-00014 University of Helsinki, Finland. Abstract
Self-avoiding walk ensembles that should converge to SLE
Tom Kennedy UC Davis, May 9, 2012 p. 1/4 Self-avoiding walk ensembles that should converge to SLE Tom Kennedy University of Arizona, MSRI Tom Kennedy UC Davis, May 9, 2012 p. 2/4 Outline Variety of ensembles
CRITICAL PERCOLATION AND CONFORMAL INVARIANCE
CRITICAL PERCOLATION AND CONFORMAL INVARIANCE STANISLAV SMIRNOV Royal Institute of Technology, Department of Mathematics, Stockholm, S10044, Sweden E-mail : stas@math.kth.se Many 2D critical lattice models
Reversibility of Some Chordal SLE(κ; ρ) Traces
Reversibility of Some Chordal SLE(κ; ρ Traces Dapeng Zhan May 14, 010 Abstract We prove that, for κ (0, 4 and ρ (κ 4/, the chordal SLE(κ; ρ trace started from (0; 0 + or (0; 0 satisfies the reversibility
Stochastic Loewner Evolution: another way of thinking about Conformal Field Theory
Stochastic Loewner Evolution: another way of thinking about Conformal Field Theory John Cardy University of Oxford October 2005 Centre for Mathematical Physics, Hamburg Outline recall some facts about
LIOUVILLE QUANTUM MULTIFRACTALITY
LIOUVILLE QUANTUM MULTIFRACTALITY Bertrand Duplantier Institut de Physique Théorique Université Paris-Saclay, France 116TH STATISTICAL MECHANICS CONFERENCE HONOREES: JOHN CARDY & SUSAN COPPERSMITH Rutgers
CFT and SLE and 2D statistical physics. Stanislav Smirnov
CFT and SLE and 2D statistical physics Stanislav Smirnov Recently much of the progress in understanding 2-dimensional critical phenomena resulted from Conformal Field Theory (last 30 years) Schramm-Loewner
The near-critical planar Ising Random Cluster model
The near-critical planar Ising Random Cluster model Gábor Pete http://www.math.bme.hu/ gabor Joint work with and Hugo Duminil-Copin (Université de Genève) Christophe Garban (ENS Lyon, CNRS) arxiv:1111.0144
This thesis studies the geometry of objects from 2-dimensional statistical physics in the continuum.
Overview This thesis studies the geometry of objects from 2-dimensional statistical physics in the continuum. Chapter 1 is an introduction to Schramm-Loewner evolutions (SLE). SLEs are the canonical family
arxiv:math-ph/ v3 4 May 2006
On Conformal Field Theory of SLE(κ, ρ arxiv:math-ph/0504057v3 4 May 006 Kalle Kytölä kalle.kytola@helsinki.fi Department of Mathematics and Statistics, P.O. Box 68 FIN-00014 University of Helsinki, Finland.
Conformal invariance and covariance of the 2d self-avoiding walk
Conformal invariance and covariance of the 2d self-avoiding walk Department of Mathematics, University of Arizona AMS Western Sectional Meeting, April 17, 2010 p.1/24 Outline Conformal invariance/covariance
CONFORMAL INVARIANCE AND 2 d STATISTICAL PHYSICS
CONFORMAL INVARIANCE AND 2 d STATISTICAL PHYSICS GREGORY F. LAWLER Abstract. A number of two-dimensional models in statistical physics are conjectured to have scaling limits at criticality that are in
Locality property and a related continuity problem for SLE and SKLE I
Locality property and a related continuity problem for SLE and SKLE I Masatoshi Fukushima (Osaka) joint work with Zhen Qing Chen (Seattle) October 20, 2015 Osaka University, Σ-hall 1 Locality property
An Introduction to Percolation
An Introduction to Percolation Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ kozdron/ Colloquium Department of Mathematics & Statistics September 28, 2007 Abstract Percolation was
On the backbone exponent
On the backbone exponent Christophe Garban Université Lyon 1 joint work with Jean-Christophe Mourrat (ENS Lyon) Cargèse, September 2016 C. Garban (univ. Lyon 1) On the backbone exponent 1 / 30 Critical
THE WORK OF WENDELIN WERNER
THE WORK OF WENDELIN WERNER International Congress of Mathematicians Madrid August 22, 2006 C. M. Newman Courant Institute of Mathematical Sciences New York University It is my pleasure to report on some
SLE and nodal lines. Eugene Bogomolny, Charles Schmit & Rémy Dubertrand. LPTMS, Orsay, France. SLE and nodal lines p.
SLE and nodal lines p. SLE and nodal lines Eugene Bogomolny, Charles Schmit & Rémy Dubertrand LPTMS, Orsay, France SLE and nodal lines p. Outline Motivations Loewner Equation Stochastic Loewner Equation
Gradient interfaces with and without disorder
Gradient interfaces with and without disorder Codina Cotar University College London September 09, 2014, Toronto Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective
Makarov s LIL for SLE
Makarov s LIL for SLE Nam-Gyu Kang Department of Mathematics, M.I.T. Workshop at IPAM, 2007 Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, 2007 1 / 24 Outline 1 Introduction and Preliminaries
On conformally invariant CLE explorations
On conformally invariant CLE explorations Wendelin Werner Hao Wu arxiv:1112.1211v2 [math.pr] 17 Dec 2012 Abstract We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles
Random planar curves Schramm-Loewner Evolution and Conformal Field Theory
Random planar curves Schramm-Loewner Evolution and Conformal Field Theory John Cardy University of Oxford WIMCS Annual Meeting December 2009 Introduction - lattice models in two dimensions and random planar
Liouville quantum gravity as a mating of trees
Liouville quantum gravity as a mating of trees Bertrand Duplantier, Jason Miller and Scott Sheffield arxiv:1409.7055v [math.pr] 9 Feb 016 Abstract There is a simple way to glue together a coupled pair
Towards conformal invariance of 2D lattice models
Towards conformal invariance of 2D lattice models Stanislav Smirnov Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising
Reflected Brownian motion in generic triangles and wedges
Stochastic Processes and their Applications 117 (2007) 539 549 www.elsevier.com/locate/spa Reflected Brownian motion in generic triangles and wedges Wouter Kager Instituut voor Theoretische Fysica, Universiteit
Numerical simulation of random curves - lecture 1
Numerical simulation of random curves - lecture 1 Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk 2008 Enrage Topical School ON GROWTH
SLE(κ, ρ ) and Conformal Field Theory arxiv:math-ph/ v1 10 Dec 2004
SLE(κ, ρ ) and Conformal Field Theory arxiv:math-ph/0412033v1 10 Dec 2004 John Cardy Rudolf Peierls Centre for Theoretical Physics 1 Keble Road, Oxford OX1 3NP, U.K. and Institute for Advanced Study, Princeton
arxiv:math/ v3 [math.pr] 27 Aug 2008
![arxiv:math/ v3 [math.pr] 27 Aug 2008](/thumbs/80/80775551.jpg "arxiv:math/ v3 [math.pr] 27 Aug 2008")
The Annals of Probability 2008, Vol. 36, No. 4, 1421 1452 DOI: 10.1214/07-AOP364 c Institute of Mathematical Statistics, 2008 arxiv:math/0211322v3 [math.pr] 27 Aug 2008 THE DIMENSION OF THE SLE CURVES
Gradient Percolation and related questions
and related questions Pierre Nolin (École Normale Supérieure & Université Paris-Sud) PhD Thesis supervised by W. Werner July 16th 2007 Introduction is a model of inhomogeneous percolation introduced by
The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution
arxiv:1408.6714v3 [math.pr] 16 Apr 2015 The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution Tom Kennedy Department of Mathematics University of Arizona Tucson, AZ 85721 email: tgk@math.arizona.edu
Uniform spanning trees and loop-erased random walks on Sierpinski graphs
Uniform spanning trees and loop-erased random walks on Sierpinski graphs Elmar Teufl Eberhard Karls Universität Tübingen 12 September 2011 joint work with Stephan Wagner Stellenbosch University Elmar Teufl
SLE 6 and CLE 6 from critical percolation
Probability, Geometry and Integrable Systems MSRI Publications Volume 55, 2008 SLE 6 and CLE 6 from critical percolation FEDERICO CAMIA AND CHARLES M. NEWMAN ABSTRACT. We review some of the recent progress
Stochastic Schramm-Loewner Evolution (SLE) from Statistical Conformal Field Theory (CFT): An Introduction for (and by) Amateurs
Denis Bernard Stochastic SLE from Statistical CFT 1 Stochastic Schramm-Loewner Evolution (SLE) from Statistical Conformal Field Theory (CFT): An Introduction for (and by) Amateurs Denis Bernard Chern-Simons
Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
SLE for Theoretical Physicists
arxiv:cond-mat/0503313v2 [cond-mat.stat-mech] 2 May 2005 SLE for Theoretical Physicists John Cardy Rudolf Peierls Centre for Theoretical Physics 1 Keble Road, Oxford OX1 3NP, U.K. and All Souls College,
Brownian Motion with Darning applied to KL and BF equations for planar slit domains
Brownian Motion with Darning applied to KL and BF equations for planar slit domains Masatoshi Fukushima with Z.-Q. Chen and S. Rohde September 26, 2012 at Okayama University Stochastic Analysis and Applications
Loewner Evolution. Maps and Shapes in two Dimensions. presented by. Leo P. Kadanoff University of Chicago.
Loewner Evolution Maps and Shapes in two Dimensions presented by Leo P. Kadanoff University of Chicago e-mail: LeoP@UChicago.edu coworkers Ilya Gruzberg, Bernard Nienhuis, Isabelle Claus, Wouter Kager,
Poisson point processes, excursions and stable processes in two-dimensional structures
Stochastic Processes and their Applications 120 (2010) 750 766 www.elsevier.com/locate/spa Poisson point processes, excursions and stable processes in two-dimensional structures Wendelin Werner Université
Simple CLE in Doubly Connected Domains
Simple CLE in Doubly Connected Domains Scott Sheffield, Samuel S. Watson and Hao Wu arxiv:40.632v2 [math.pr] 5 Nov 205 Abstract We study Conformal Loop Ensemble (CLE κ ) in doubly connected domains: annuli,
Conformal blocks in nonrational CFTs with c 1
Conformal blocks in nonrational CFTs with c 1 Eveliina Peltola Université de Genève Section de Mathématiques < eveliina.peltola@unige.ch > March 15th 2018 Based on various joint works with Steven M. Flores,
The Length of an SLE - Monte Carlo Studies
The Length of an SLE - Monte Carlo Studies Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ tgk Stochastic Geometry and Field Theory, KITP,
Convergence of 2D critical percolation to SLE 6
Convergence of 2D critical percolation to SLE 6 Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ kozdron/ Lecture given at the Mathematisches Forschungsinstitut Oberwolfach (MFO) during
APPLICATIONS of QUANTUM GROUPS to CONFORMALLY INVARIANT RANDOM GEOMETRY
APPLICATIONS of QUANTUM GROUPS to CONFORMALLY INVARIANT RANDOM GEOMETRY Eveliina Peltola Academic dissertation To be presented for public examination with the permission of the Faculty of Science of the
1 Introduction. 2 Simulations involving conformal maps. 2.1 Lowener equation crash course
June 1, 2008. This is very preliminary version of these notes. In particular the references are grossly incomplete. You should check my website to see if a newer version of the notes is available. 1 Introduction
Random curves, scaling limits and Loewner evolutions
Random curves, scaling limits and Loewner evolutions Antti Kemppainen Stanislav Smirnov June 15, 2015 arxiv:1212.6215v3 [math-ph] 12 Jun 2015 Abstract In this paper, we provide a framework of estimates
Natural parametrization of percolation interface and pivotal points
Natural parametrization of percolation interface and pivotal points Nina Holden Xinyi Li Xin Sun arxiv:1804.07286v2 [math.pr] 3 Apr 2019 Abstract We prove that the interface of critical site percolation
arxiv:math/ v1 [math.pr] 27 Mar 2003
![arxiv:math/ v1 [math.pr] 27 Mar 2003](/thumbs/74/70647649.jpg "arxiv:math/ v1 [math.pr] 27 Mar 2003")
1 arxiv:math/0303354v1 [math.pr] 27 Mar 2003 Random planar curves and Schramm-Loewner evolutions Lecture Notes from the 2002 Saint-Flour summer school (final version) Wendelin Werner Université Paris-Sud
Liouville quantum gravity and the Brownian map I: The QLE(8/3, 0) metric
Liouville quantum gravity and the Brownian map I: The QLE(8/3, 0) metric arxiv:1507.00719v2 [math.pr] 27 Feb 2016 Jason Miller and Scott Sheffield Abstract Liouville quantum gravity (LQG) and the Brownian
3 Statistical physics models
3 Statistical physics models 3.1 Percolation In Werner s St. Flour lectures he discusses percolation in the first section and then in more detail in section 10. In the article by Kager and Nienhuis percolation
Mathematical Research Letters 8, (2001) THE DIMENSION OF THE PLANAR BROWNIAN FRONTIER IS 4/3
Mathematical Research Letters 8, 401 411 (2001) THE DIMENSION OF THE PLANAR BROWNIAN FRONTIER IS 4/3 Gregory F. Lawler 1, Oded Schramm 2, and Wendelin Werner 3 1. Introduction The purpose of this note
arxiv: v3 [math.pr] 27 Nov 2018
![arxiv: v3 [math.pr] 27 Nov 2018](/thumbs/93/113906067.jpg "arxiv: v3 [math.pr] 27 Nov 2018")
DIMERS AND IMAGINARY GEOMETRY By Nathanaël Berestycki, Benoît Laslier and Gourab Ray Universität Wien, Université Paris Diderot and University of Victoria arxiv:1603.09740v3 [math.pr] 27 Nov 2018 We show
Extremal process associated with 2D discrete Gaussian Free Field
Extremal process associated with 2D discrete Gaussian Free Field Marek Biskup (UCLA) Based on joint work with O. Louidor Plan Prelude about random fields blame Eviatar! DGFF: definitions, level sets, maximum
In terms of measures: Exercise 1. Existence of a Gaussian process: Theorem 2. Remark 3.
1. GAUSSIAN PROCESSES A Gaussian process on a set T is a collection of random variables X =(X t ) t T on a common probability space such that for any n 1 and any t 1,...,t n T, the vector (X(t 1 ),...,X(t
Testing for SLE using the driving process
Testing for SLE using the driving process Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk Testing for SLE, 13rd Itzykson Conference
Random curves, scaling limits and Loewner evolutions. Kemppainen, Antti
https://helda.helsinki.fi Random curves, scaling limits and Loewner evolutions Kemppainen, Antti 2017-03 Kemppainen, A & Smirnov, S 2017, ' Random curves, scaling limits and Loewner evolutions ', Annals
arxiv:math-ph/ v2 14 Mar 2003
Proceedings of Symposia in Pure Mathematics Amérique, Ô ma Norvège! Dominique Fourcade Le sujet monotype P.O.L. (1997) arxiv:math-ph/0303034v2 14 Mar 2003 CONFORMAL FRACTAL GEOMETRY & BOUNDARY QUANTUM
Nonparametric Drift Estimation for Stochastic Differential Equations
Nonparametric Drift Estimation for Stochastic Differential Equations Gareth Roberts 1 Department of Statistics University of Warwick Brazilian Bayesian meeting, March 2010 Joint work with O. Papaspiliopoulos,
arxiv: v2 [math.pr] 5 May 2015
![arxiv: v2 [math.pr] 5 May 2015](/thumbs/90/103269355.jpg "arxiv: v2 [math.pr] 5 May 2015")
An SLE 2 loop measure Stéphane Benoist Julien Dubédat June 25, 218 arxiv:145.788v2 [math.pr] 5 May 215 Abstract There is an essentially unique way to associate to any Riemann surface a measure on its simple
Topics on the two-dimensional Gaussian Free Field. Wendelin Werner
Topics on the two-dimensional Gaussian Free Field Wendelin Werner Lecture Notes (preliminary version 31/12/14) of the topics course at ETH Zürich, September-December 2014 Contents 1 A few things about
Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra
Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra kalle.kytola@aalto.fi Department of Mathematics and Systems Analysis, Aalto University joint work with March 5,
Liouville quantum gravity spheres as matings of finite-diameter trees
Liouville quantum gravity spheres as matings of finite-diameter trees arxiv:1506.03804v3 [math.pr] 10 Apr 2017 Jason Miller and Scott Sheffield Abstract We show that the unit area Liouville quantum gravity
ALTERNATING ARM EXPONENTS FOR THE CRITICAL PLANAR ISING MODEL. By Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China
Submitted to the Annals of Probability arxiv: arxiv:1605.00985 ALTERNATING ARM EXPONENTS FOR THE CRITICAL PLANAR ISING MODEL By Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China We derive
Extrema of discrete 2D Gaussian Free Field and Liouville quantum gravity
Extrema of discrete 2D Gaussian Free Field and Liouville quantum gravity Marek Biskup (UCLA) Joint work with Oren Louidor (Technion, Haifa) Discrete Gaussian Free Field (DGFF) D R d (or C in d = 2) bounded,
Gaussian Fields and Percolation
Gaussian Fields and Percolation Dmitry Beliaev Mathematical Institute University of Oxford RANDOM WAVES IN OXFORD 18 June 2018 Berry s conjecture In 1977 M. Berry conjectured that high energy eigenfunctions
Stochastic Loewner evolution in doubly connected domains
arxiv:math/0310350v3 [math.pr] 5 Sep 2004 Stochastic Loewner evolution in doubly connected domains Dapeng Zhan March, 2004 Abstract This paper introduces the annulus SLE κ processes in doubly connected
Interfaces between Probability and Geometry
Interfaces between Probability and Geometry (Prospects in Mathematics Durham, 15 December 2007) Wilfrid Kendall w.s.kendall@warwick.ac.uk Department of Statistics, University of Warwick Introduction Brownian
Scaling limits of anisotropic random growth models
Scaling limits of anisotropic random growth models Amanda Turner Department of Mathematics and Statistics Lancaster University (Joint work with Fredrik Johansson Viklund and Alan Sola) Overview 1 Generalised
Gradient Gibbs measures with disorder
Gradient Gibbs measures with disorder Codina Cotar University College London April 16, 2015, Providence Partly based on joint works with Christof Külske Outline 1 The model 2 Questions 3 Known results
LOGARITHMIC COEFFICIENTS AND GENERALIZED MULTIFRACTALITY OF WHOLE-PLANE SLE
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
A Tour of Spin Glasses and Their Geometry
A Tour of Spin Glasses and Their Geometry P. Le Doussal, D. Bernard, LPTENS A. Alan Middleton, Syracuse University Support from NSF, ANR IPAM: Random Shapes - Workshop I 26 March, 2007 Goals Real experiment
Harmonic Functions and Brownian motion
Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F
2 Wendelin Werner had been also predicted by theoretical physics). We then show how all these problems are mathematically related, and, in particular,
Critical exponents, conformal invariance and planar Brownian motion Wendelin Werner Abstract. In this review paper, we rst discuss some open problems related to two-dimensional self-avoiding paths and
Obliquely Reflected Brownian motions (ORBMs) in Non-Smooth Domains
Obliquely Reflected Brownian motions (ORBMs) in Non-Smooth Domains Kavita Ramanan, Brown University Frontier Probability Days, U. of Arizona, May 2014 Why Study Obliquely Reflected Diffusions? Applications
arxiv:math-ph/ v1 23 Aug 2006
arxiv:math-ph/0608053v1 23 Aug 2006 CONFORMAL RANDOM GEOMETRY Bertrand Duplantier Service de Physique Théorique, Orme des Merisiers, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, FRANCE 1 Photo: width 7.5cm
Fast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
Continuous LERW started from interior points
Stochastic Processes and their Applications 120 (2010) 1267 1316 www.elsevier.com/locate/spa Continuous LERW started from interior points Dapeng Zhan Department of Mathematics, Michigan State University,
Densities for the Navier Stokes equations with noise
Densities for the Navier Stokes equations with noise Marco Romito Università di Pisa Universitat de Barcelona March 25, 2015 Summary 1 Introduction & motivations 2 Malliavin calculus 3 Besov bounds 4 Other
Minkowski content of the intersection of a Schramm-Loewner evolution (SLE) curve with the real line
Submitted to Journal of the Mathematical Society of Japan Minkowski content of the intersection of a Schramm-Loewner evolution (SLE) curve with the real line By Gregory F. Lawler Abstract. The Schramm-Loewner
3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
arxiv: v2 [math-ph] 21 Apr 2017
![arxiv: v2 [math-ph] 21 Apr 2017](/thumbs/94/121660484.jpg "arxiv: v2 [math-ph] 21 Apr 2017")
LOGARITHMIC COEFFICIENTS AND GENERALIZED MULTIFRACTALITY OF WHOLE-PLANE SLE BERTRAND DUPLANTIER 1), XUAN HIEU HO ), THANH BINH LE ), AND MICHEL ZINSMEISTER ) arxiv:1504.05570v [math-ph] 1 Apr 017 Abstract.
CONVERGENCE RATES FOR LOOP-ERASED RANDOM WALK AND OTHER LOEWNER CURVES. BY FREDRIK JOHANSSON VIKLUND 1 Columbia University
The Annals of Probability 2015, Vol. 43, No. 1, 119 165 DOI: 10.1214/13-AOP872 Institute of Mathematical Statistics, 2015 CONVERGENCE RATES FOR LOOP-ERASED RANDOM WALK AND OTHER LOEWNER CURVES BY FREDRIK
PERCOLATION AND COARSE CONFORMAL UNIFORMIZATION. 1. Introduction
PERCOLATION AND COARSE CONFORMAL UNIFORMIZATION ITAI BENJAMINI Abstract. We formulate conjectures regarding percolation on planar triangulations suggested by assuming (quasi) invariance under coarse conformal
Extreme values of two-dimensional discrete Gaussian Free Field
Extreme values of two-dimensional discrete Gaussian Free Field Marek Biskup (UCLA) Joint work with Oren Louidor (Technion, Haifa) Midwest Probability Colloquium, Evanston, October 9-10, 2015 My collaborator
FRACTAL AND MULTIFRACTAL PROPERTIES OF SLE
Clay Mathematics Proceedings FRACTAL AND MULTIFRACTAL PROPERTIES OF SLE Gregory F. Lawler Introduction This is a slightly expanded version of my lectures at the 2010 Clay Mathematics Institute summer (winter)