Imaginary multiplicative chaos and the XOR-Ising model Janne Junnila - - PowerPoint PPT Presentation

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Imaginary multiplicative chaos and the XOR-Ising model

Janne Junnila (EPFL) joint work with Eero Saksman (University of Helsinki) Christian Webb (Aalto University)

Mathematical Physics Seminar (UNIGE), April 8th 2019

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

A bit of history

The study of multiplicative chaos traces back to the works of Høegh-Krohn and Mandelbrot in the early 70s. Mandelbrot proposed to improve Kolmogorov’s log-normal model

• f energy dissipation in turbulence by using random measures of

the form

𝑒𝜈(𝑦) ≔ 𝑓𝛿𝑌(𝑦)− 𝛿2

2 𝔽𝑌(𝑦)2 𝑒𝑦,

where 𝑌 is a log-correlated Gaussian field on some domain 𝑉 ⊂ ℝ𝑒 and 𝛿 > 0 is a parameter. The model was revisited and rigorously studied by Kahane in 1985 who coined the term Gaussian multiplicative chaos (GMC). There has been a renessaince of interest in the last 10 years: connections to Liouville quantum gravity, SLE, random matrices etc.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Log-correlated Gaussian fields

A Gaussian generalized function with covariance (kernel) of the form

"𝔽𝑌(𝑦)𝑌(𝑧)" = log 1 |𝑦 − 𝑧| + 𝑕(𝑦, 𝑧).

We assume that 𝑕 is integrable, continuous and bounded from above.

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Figure: A simulation of 2D Gaussian Free Field

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Rigorous definition of GMC measures

The GMC measure 𝜈 is typically defined by approximating the field

𝑌 with regular fields 𝑌𝜁 and taking a limit as 𝜁 → 0 of approximating

measures

𝑒𝜈𝜁(𝑦) ≔ 𝑓𝛿𝑌𝜁(𝑦)− 𝛿2

2 𝔽𝑌𝜁(𝑦)2 𝑒𝑦.

Easy case: 𝑀2-bounded martingales

If (𝑌𝜁)𝜁>0 is a martingale in 𝜁, convergence to a non-trivial limit is easily obtained when 𝛿 ∈ (0, √𝑒) by checking that we have boundedness in 𝑀2(𝛻):

𝔽 |∫

𝑉

𝑔(𝑦)𝑒𝜈𝜁(𝑦)|

2

= ∫

𝑉

∫

𝑉

𝑔(𝑦)𝑔(𝑧)𝔽𝑓𝛿𝑌𝜁(𝑦)+𝛿𝑌𝜁(𝑧)− 𝛿2

2 𝔽𝑌𝜁(𝑦)2− 𝛿2 2 𝔽𝑌𝜁(𝑧)2 𝑒𝑦 𝑒𝑧

≲ ‖𝑔‖2

∞ ∫ 𝑉

∫

𝑉

|𝑦 − 𝑧|−𝛿2 𝑒𝑦 𝑒𝑧 < ∞

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Properties

• Convergence holds for 𝛿 ∈ (0, √2𝑒).
• Moments: 𝔽| ∫

𝐿 𝑒𝜈(𝑦)|𝑞 < ∞ if and only if 𝑞 < 2𝑒 𝛿2 .

• Support: 𝜈 gives full measure to the set

{𝑦 ∈ 𝑉 ∶ lim

𝜁→0

𝑌𝜁(𝑦) 𝔽𝑌𝜁(𝑦)2 = 𝛿},

which has Hausdorff dimension equal to 𝑒 − 𝛿2

2 .

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Complex multiplicative chaos

One can also define GMC distributions for complex values of 𝛿.

ℜ(𝛿) ℑ(𝛿) √𝑒 −√𝑒 −√2𝑒 √2𝑒

Figure: The subcritical regime for 𝛿 in the complex plane.

We will from now on focus on the case 𝛿 = 𝑗𝛾, with 𝛾 ∈ (0, √𝑒).

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Existence

Let 𝜒∶ ℝ𝑒 → ℝ be a smooth mollifier, set 𝜒𝜁(𝑦) = 1

𝜁𝑒 𝜒( 𝑦 𝜁 ) and define

the approximating fields 𝑌𝜁 = 𝑌 ∗ 𝜒𝜁. As we are inside the 𝑀2-phase, one can show that the functions

𝜈𝜁 ≔ 𝑓𝑗𝛾𝑌𝜁(𝑦)+ 𝛾2

2 𝔽𝑌𝜁(𝑦)2

form a Cauchy sequence in 𝑀2(𝛻) as elements of 𝐼𝑡(ℝ𝑒) for

𝑡 < −𝑒/2 and consequently obtain convergence in probability to a

random distribution 𝜈 ∈ 𝐼𝑡(ℝ𝑒). Similar 𝑀2-computations also show that the limit does not depend on the choice of 𝜒. More generally one can prove uniqueness and convergence for a wider class of so called standard approximations.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Moments

All (mixed) moments of 𝜈(𝑔) are finite and

𝔽|𝜈(𝑔)|2𝑂 ≤ ‖𝑔‖2𝑂

∞ 𝐷𝑂𝑂

𝛾2 𝑒 𝑂.

In particular, the moments determine the distribution of 𝜈(𝑔).

The case of pure logarithm covariance

If 𝔽𝑌(𝑦)𝑌(𝑧) = log

1 |𝑦−𝑧|, a straightforward computations yields

𝔽|𝜈(1)|2𝑂 = ∫

𝑉2𝑂

∏1≤𝑗<𝑘≤𝑂 |𝑦𝑗 − 𝑦𝑘|𝛾2 ∏1≤𝑗<𝑘≤𝑂 |𝑧𝑘 − 𝑧𝑘|𝛾2 ∏1≤𝑗,𝑘≤𝑂 |𝑦𝑗 − 𝑧𝑘|𝛾2 .

Estimating this was done by Gunson and Panta in 1977 in two dimensions, but for other dimensions and more general covariances some extra work is needed.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Moments for general covariances

In general the formula for 2𝑂th moment is

𝔽|𝜈(1)|2𝑂 = ∫

𝑉2𝑂 𝑓−𝛾2 ∑1≤𝑗<𝑘≤𝑂 𝐷(𝑦𝑗,𝑦𝑘)−𝛾2 ∑1≤𝑗<𝑘≤𝑂 𝐷(𝑧𝑗,𝑧𝑘)+𝛾2 ∑1≤𝑗,𝑘≤𝑂 𝐷(𝑦𝑗,𝑧𝑘).

Naive approach

If one simply assumes that the 𝑕-term in the covariance of 𝑌 is bounded, then one could bound the exponent by (𝛾2 times)

− ∑

1≤𝑗<𝑘≤𝑂

log 1 |𝑦𝑗 − 𝑦𝑘| − ∑

1≤𝑗<𝑘≤𝑂

log 1 |𝑧𝑗 − 𝑧𝑘| + ∑

1≤𝑗,𝑘≤𝑂

log 1 |𝑦𝑗 − 𝑧𝑘| +𝐷𝑂2

for some constant 𝐷 > 0 and in this way reduce to the pure-logarithm case.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Onsager inequalities

Onsager inequalities provide a better bound for the exponent. If we let 𝑟1, … , 𝑟2𝑂 ∈ {−1, 1}, then we have

− ∑

1≤𝑗<𝑘≤2𝑂

𝑟𝑗𝑟𝑘𝐷(𝑦𝑗, 𝑦𝑘) ≤ 1 2

2𝑂

∑

𝑗=1

log 1

1 2 min𝑘≠𝑗 |𝑦𝑗 − 𝑦𝑘|

+ 𝐷𝑂.

The integral of the exponential of the RHS can then be estimated using a combinatorial argument already appearing in the Gunson and Panta paper. We have a couple of versions of this inequality, depending on what regularity one assumes from 𝑕. Either

• 𝑒 = 2 and 𝑕 ∈ 𝐷2(𝑉 × 𝑉) (get Onsager on any compact subset

𝐿 ⊂ 𝑉); or

• 𝑒 ≥ 1 arbitrary 𝑕 ∈ 𝐼𝑒+𝜁

𝑚𝑝𝑑 (𝑉 × 𝑉) for some 𝜁 > 0 (get Onsager

locally on small enough balls); or

• 𝑒 = 2 and 𝑌 is the GFF (get Onsager globally in 𝑉)

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Interlude: Besov spaces

Besov spaces 𝐶𝑡

𝑞,𝑟(ℝ𝑒) are Banach spaces of (generalized) functions

in ℝ𝑒 parametrised by three parameters 𝑡 ∈ ℝ, 1 ≤ 𝑞, 𝑟 ≤ ∞. The Besov-norm is defined by

‖𝑔‖𝐶𝑡

𝑞,𝑟(ℝ𝑒) ≔ ‖(2𝑙𝑡‖𝜚𝑙 ∗ 𝑔‖𝑀𝑞(ℝ𝑒))∞

𝑙=0‖ℓ𝑟(ℕ),

where 𝜚𝑙 ∈ S (ℝ𝑒), 𝜚𝑙(𝑦) ≔ 2(𝑙−1)𝑒𝜚1(2𝑙−1𝑦) for 𝑙 ≥ 2,

supp ̂ 𝜚0 ⊂ 𝐶(0, 2), supp ̂ 𝜚1 ⊂ 𝐶(0, 4) ⧵ 𝐶(0, 1) and ∑∞

𝑙=0

̂ 𝜚𝑙(𝜊) ≡ 1.

The Besov spaces include many common function spaces, and in particular

• 𝐶𝑡

2,2(ℝ𝑒) = 𝐼𝑡(ℝ𝑒)

• 𝐶𝑡

∞,∞(ℝ𝑒) = 𝐷𝑡(ℝ𝑒) (at least for 𝑡 ∈ (0, 1))

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Regularity of imaginary chaos

• 𝜈 is almost surely not a complex measure (it has infinite total

variation).

• As a random distribution we have 𝜈 ∈ 𝐶𝑡

𝑞,𝑟,𝑚𝑝𝑑(𝑉) a.s. when

𝑡 < − 𝛾2

2 and 𝜈 ∉ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝑉) when 𝑡 > − 𝛾2 2 .

• log ℙ[|𝜈(𝑔)| > 𝜇) behaves approximately like 𝜇

2𝑒 𝛾2 as 𝜇 → ∞.

• As 𝛾 → √𝑒, we converge in law to a weighted complex white

noise:

√𝑒 − 𝛾2 |𝑇𝑒−1| 𝜈𝛾(𝑦) → 𝑓

𝛾2 2 𝑕(𝑦,𝑦)𝑋(𝑒𝑦)

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Proof ideas

To show that 𝜈 is a.s. not a complex measure it is enough to show that there a.s. exists a sequence 𝑔

𝑙 ∈ 𝐷𝑑(𝑉) with ‖𝑔 𝑙‖∞ ≤ 1 and

|𝜈(𝑔

𝑙)| → ∞. A suitable sequence is given by 𝜔(𝑦)𝑓−𝑗𝛾𝑌𝜁𝑙(𝑦), where

𝜔 ∈ 𝐷∞

𝑑 (𝑉). We show that (𝔽𝜈(𝑔

𝑙))2

𝔽|𝜈(𝑔

𝑙)|2 → 1 and use the Paley–Zygmund

inequality

ℙ[|𝜈(𝑔

𝑙)| ≥ 𝜄𝔽|𝜈(𝑔 𝑙)|] ≥ (1 − 𝜄)2 (𝔽𝜈(𝑔 𝑙))2

𝔽|𝜈(𝑔

𝑙)|2

with 𝜄 = (𝔽|𝜈(𝑔

𝑙)|)−𝜁.

To show that 𝜈 ∈ 𝐶𝑡

𝑞,𝑟,𝑚𝑝𝑑(𝑉), 𝑡 < − 𝛾2 2 , it is enough to focus on the

case 𝑞 = 𝑟 = 2𝑂 and show that 𝔽‖𝜔𝜈‖2𝑂

𝐶𝑡

2𝑂,2𝑂(ℝ𝑒) < ∞.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

A small note on universality

We mentioned before that the chaos does not typically depend on the approximation used. We can also show another type of universality: Instead of looking at 𝑓𝑗𝛾𝑌(𝑦)+ 𝛾2

2 𝔽𝑌(𝑦)2, one can look at

𝐼(𝛾𝑌(𝑦))𝑓

𝛾2 2 𝔽𝑌(𝑦)2, where 𝐼 is a 2𝜌-periodic function with

absolutely converging Fourier series and mean zero. One then gets chaos corresponding to

̂ 𝐼1𝑓𝑗𝛾𝑌(𝑦)+ 𝛾2

2 𝔽𝑌(𝑦)2 +

̂ 𝐼−1𝑓−𝑗𝛾𝑌(𝑦)+ 𝛾2

2 𝔽𝑌(𝑦)2.

In particular, if 𝐼 is an even function it gives rise to just a constant times the real part of the imaginary chaos.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Ising model

• + (red) or − (blue) spins

𝜏𝜀(𝑔) on faces 𝑔 of a square

grid of side length 𝜀 approximating a domain 𝑉

• +-boundary condition
• probability of configuration

∝ 𝑓𝛾 ∑𝑏∼𝑐 𝜏𝜀(𝑏)𝜏𝜀(𝑐)

• 𝛾 = 𝛾𝑑 = log(1+√2)

2

• We extend the definition to

the whole domain 𝑉 by letting 𝜏𝜀(𝑦) be constant ±1

• n each face.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Correlation functions

It has been shown by Chelkak, Hongler and Izyurov that the 𝑜-point spin correlation functions have the following limit:

lim

𝜀→0+ 𝜀− 𝑜

8 𝔽

𝑜

∏

𝑘=1

𝜏𝜀(𝑦𝑘) = C 𝑜

𝑜

∏

𝑘=1

( |𝜒′(𝑦𝑘)| 2ℑ(𝜒(𝑦𝑘)))

1/8

× (2−𝑜/2 ∑

𝜈∈{−1,1}𝑜

∏

1≤𝑙<𝑛≤𝑜

|𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) 𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) |

𝜈𝑙𝜈𝑛 2

)

1/2

,

where 𝜒∶ 𝑉 → ℍ is a conformal bijection.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

XOR-Ising

XOR-Ising model is obtained by taking two independent Ising spin models 𝜏𝜀(𝑦) and ̃

𝜏𝜀(𝑦) and multiplying their spins together to get

another spin model 𝑇𝜀(𝑦) ≔ 𝜏𝜀(𝑦) ̃

𝜏𝜀(𝑦).

The correlation functions of 𝑇𝜀(𝑦) are the squares of the correlation functions of 𝜏𝜀(𝑦) and in the limit they coincide with the correlation functions of

C 2(2|𝜒′(𝑦)| ℑ(𝜒(𝑦)) )

1/4

cos( 1 √2𝑌(𝑦))

where cos( 1

√2𝑌(𝑦)) stands for the real part of the imaginary chaos

𝜈1/√2(𝑦) with 𝑌 being the zero-boundary GFF on 𝑉.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Convergence of XOR-Ising to cos(𝑌(𝑦)/√2)

The claim is that for any test function 𝑔 ∈ 𝐷∞

𝑑 (𝑉) we have

∫ 𝑔(𝑦)𝜀−1/4𝑇𝜀(𝑦) 𝑒𝑦

𝑒

→ ∫ 𝑔(𝑦)C 2(2|𝜒′(𝑦)| ℑ(𝜒(𝑦)) )

1/4

cos( 1 √2𝑌(𝑦)) 𝑒𝑦.

The idea is to use the method of moments. This works as we have shown that the moments determine the distribution of the imaginary chaos. Formally we have this convergence because of the pointwise convergence of the correlation functions, but we still need to justify the use of dominated convergence theorem in order to conclude.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Onsager-type inequality for XOR-Ising

To justify dominated convergence we use the (square of the) following variant of Onsager’s inequality for Ising model:

𝜀−𝑜/8𝔽𝜏(𝑦1) … 𝜏(𝑦𝑜) ≤ 𝐷𝑜

𝑜

∏

𝑗=1

(min

𝑘≠𝑗 |𝑦𝑗 − 𝑦𝑘|)−1/8

Here 𝐷 > 0 is a constant and the inequality holds for 𝑦𝑗 ∈ 𝐿 where 𝐿 is a compact subset of 𝑉. (The constant 𝐷 may depend on 𝐿.)

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Interlude: FK-Ising model

Consider a random cluster model on the dual lattice, where each configuration is assigned a probability proportional to

𝑞#open edges

𝑑

(1 − 𝑞𝑑)#closed edges2#open clusters,

where 𝑞𝑑 =

√2 1+√2.

Edwards–Sokal coupling

One can sample a critical Ising configuration with +-boundary condition by first sampling the random cluster model and then choosing a spin independently for each open cluster with probability 1/2 for either + or −, except for the boundary component for which one always chooses +.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Sketch of the proof of the “Ising-Onsager”

𝔽[𝜏(𝑔

1) … 𝜏(𝑔 𝑜)]

= 𝔽[𝔽[𝜏(𝑔

1) … 𝜏(𝑔 𝑜)|RC model]]

= 𝔽[

𝑜

∏

𝑗=1

𝟚{| Cl(𝑔

𝑗)| is even or Cl(𝑔 𝑗)↭𝜖𝑉}]

≤ 𝔽[

𝑜

∏

𝑗=1

𝟚{𝑔

𝑗↭𝜖𝐶𝑗}]

≤

𝑜

∏

𝑗=1

ℙ+

𝐶𝑗[𝑔 𝑗 ↭ 𝜖𝐶𝑗]

=

𝑜

∏

𝑗=1

𝔽+

𝐶𝑗[𝜏(𝑔 𝑗)]

≤ 𝐷𝑜

𝑜

∏

𝑗=1

(min

𝑘≠𝑗 𝑒(𝑔 𝑗, 𝑔 𝑘) ∧ 𝑒(𝑔 𝑗, 𝜖𝑉))−1/8.

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

Simulation of the scaling limit of XOR-Ising using GFF

−4 −2 2 4

[ March 1, 2018 at 18:04 – classicthesis version 0.1 ]

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

• D. Chelkak, C. Hongler, and K. Izyurov. “Conformal invariance of spin

correlations in the planar Ising model”. In: Annals of mathematics (2015),

• pp. 1087–1138.
• M. Furlan and J.-C. Mourrat. “A tightness criterion for random fields, with

application to the Ising model”. In: Electronic Journal of Probability 22 (2017).

• J. Gunson and L. S. Panta. “Two-dimensional neutral Coulomb gas”. In:

Communications in Mathematical Physics 52.3 (1977), pp. 295–304.

• R. Høegh-Krohn. “A general class of quantum fields without cut-offs in

two space-time dimensions”. In: Communications in Mathematical Physics 21.3 (1971), pp. 244–255.

• J. Junnila, E. Saksman, and C. Webb. “Decompositions of log-correlated

fields with applications”. In: arXiv:1808.06838 (2018).

• J. Junnila, E. Saksman, and C. Webb. “Imaginary multiplicative chaos:

Moments, regularity and connections to the Ising model”. In: arXiv:1806.02118 (2018).

Introduction to Gaussian multiplicative chaos Imaginary multiplicative chaos XOR-Ising model References

J.–P. Kahane. “Sur le chaos multiplicatif”. In: Comptes rendus de l’Académie des sciences. Série 1, Mathématique 301.6 (1985), pp. 329–332.

• B. Mandelbrot. “Possible refinement of the lognormal hypothesis

concerning the distribution of energy dissipation in intermittent turbulence”. In: Statistical Models and Turbulence. Berlin, Heidelberg: Springer, 1972, pp. 333–351.