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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011 Noise, sign problems, and statistics arXiv:1106.0073 [hep-lat] Michael Endres, D.K., Jong-Wan Lee, Amy Nicholson ...& work in progress TYPE THE THREE WORDS L i s t e n i n g t o E N O I S Listening to NOISE

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Page 1: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Noise, sign problems, and statisticsarXiv:1106.0073 [hep-lat]Michael Endres, D.K., Jong-Wan Lee, Amy Nicholson...& work in progress

CAPTCHA: Telling Humans and Computers Apart Automatically

A CAPTCHA is a program that protects websites against bots by generating and grading tests thathumans can pass but current computer programs cannot. For example, humans can read distortedtext as the one shown below, but current computer programs can't:

The term CAPTCHA (for Completely Automated Public Turing Test To Tell Computers and HumansApart) was coined in 2000 by Luis von Ahn, Manuel Blum, Nicholas Hopper and John Langford ofCarnegie Mellon University.

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Our new site, GWAP.com contains many addictive games that helpcomputers learn to think more like humans. You play the games, computersget smarter!

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TYPE THE THREE WORDS

Listening to ENOISListening to NOISE

Page 2: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Physics motivation: can’t we get beyond this cartoon??

Sign problem!exotic phases

?

This talk:

• From sign problem to noise

• Surprisingly universal features of noise

• Can these features be used to tame the noise?

Page 3: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

∆µ =�mN

3− mπ

2

• physics happens for μ≥ mN/3...• ...but sign problem starts at μ=mπ/2 ! P.E.

Gibbs, 198

6

Explanation (2-flavor QCD):

The “sign’’ problem in the grand canonical approach: Det(D+μγ0) complex/

|Det(D+μγ0)| ≈ isospin chemical potential

Role of phase: eliminate pion condensate for μ≥mπ/2! / K. S

plittorff

J. Verbaa

rschot, 2

006

mπ/2 μ

�π�

�nB�sign problem

mN/3

Page 4: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Canonical approach?Compute correlator of N quarks with μ=0No sign problem...but now a noise problem

T C(A)

nucleon correlatorsignal: ∼ e−mN T

C†(A)C(A)

3q3q, 3q

_

noise: ∼ 1√Nconf.

e−32 mπT

∼�

Nconf.e−3T(mN

3 −mπ2 )signal

noise____ Same factor as

grand canonical

Parisi, Lepage1980’s

Page 5: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

0 10 20 30 40 50!0.04

!0.02

0.00

0.02

0.04

t!bt

b tE

Triton B.E. S. R. Beane et al. (NPLQCD), Phys. Rev. D 80, 074501 (2009) (mπ = 390 MeV)

excited states

plateau

NOISE

EXAMPLE:

Page 6: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Actual QCD data

Plotted: −1t

lnσ(t)x̄(t)

∼ A

�mN −

32mπ

2�

mN −32mπ

�mN −

32mπ

pp

p

Conclusion: Parisi/Lepage = qualitative estimate of noise problem

Page 7: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Am I going to be in a light pion? Or a heavy nucleon?

Don’t know!

Play safe: assume pion. Propagator:

If nucleon, cancellations between configurations to:

Think like a quark in a single gauge configurationgrand canonical

XXX

XXXcanonical

∼ e−(mπ/2)T

∼ e−(mN /3)T

Page 8: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Am I going to be in a light pion? Or a heavy nucleon?

Don’t know!

Play safe: assume pion. Propagator:

If nucleon, cancellations between configurations to:

Think like a quark in a single gauge configurationgrand canonical

XXX

XXXcanonical

∼ e−(mπ/2)T

∼ e−(mN /3)T

NOISE

Page 9: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Distribution of A-baryon correlators might look like:

Aexponentially small mean μ

large variance σ

∼ e−AmN T

∼ e−32 AmπT

some non-universal distribution

Page 10: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Distribution of A-baryon correlators might look like:

Aexponentially small mean μ

large variance σ

This picture is actually not right.

Look at a simpler system where we can get large statistics: unitary fermions.

∼ e−AmN T

∼ e−32 AmπT

some non-universal distribution

Page 11: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

A =4π

M

1p cot δ − ip

phaseshift

Digression: what are unitary fermions?

Nonrelativistic 2-particle scattering:

(Lattice 2011 talks by J. Drut, J.-W. Lee)

“unitary” fermions: p cot δ = 0

δ(p) =π

2

Page 12: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

A =4π

M

1p cot δ − ip

phaseshift

Digression: what are unitary fermions?

Nonrelativistic 2-particle scattering:

(Lattice 2011 talks by J. Drut, J.-W. Lee)

“unitary” fermions: p cot δ = 0

δ(p) =π

2What is unitary fermion?

Unitary fermions : Spin 1/2 fermions with attractive interactions

� Universal� Strongly interacting

� Non-relativistic conformal

No intrinsic scale except density (n)

Universal constant ξ (Bertschparameter)

E(n) = ξEfree(n)

Pairing gap : energy cost to break a pair

∆(n)

µfree(n)= constant

⇐⇒ p cot δ0 = 0 (or δ0 = π/2)

r�r�

r

a���

zero energy bound state

r0�0

Jong-Wan Lee (UW Physics) Lattice calculation for unitary fermions June 18, 2010 4 / 16

Zero-range potential Zero-energy bound state

A strongly-coupled conformal system

Studied experimentally with dilute trapped atoms @ Feshbach resonance

Exhibits superfluidity

(JILA, MIT, Innsbruck)

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

•Less severe sign problem than QCD; no gauge symmetry; nonrelativistic (quenched)

•Have simulated up to N=70 fermions on 143 x 64 lattice

•~1% accuracy in energies

• up to 2 billion configurations for auxiliary field

Lattice model:

Nonrelativistic fermions, μ=0

Short-range momentum-dependent 4-fermion interaction induced by auxiliary scalar field

Interaction tuned to conformal fixed pt.

M. Endres, D.K., J.W. Lee, A. Nicholson, arXiv:1106.5725 [hep-lat]

ϕ

Page 14: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Simulation:

ϕϕ lives on time links (couples to ψ*ψ)

Generate an ensemble of random ϕ fields, compute average of an N-particle correlator CN(T;ϕ) from t=0 to t=T

Extract ground state energy:

EN = limT→∞

�− 1

T ln�CN (T,φ)�φ�

Plotvs T, look for “plateau”

− 1T ln�CN (T,φ)�φ

Page 15: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

⇓⇑

t

x

⇓⇑

y

(1 + gφ(x, t))

Correlators are products of many transfer matrices in background random Φ

Page 16: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Effective mass plot with standard technique

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

meff���E

Free

40 M configs

23

Give up?

Example of conventional effective mass plot

N= 46 fermionsL=12

Page 17: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Effective mass plot with standard technique

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

meff���E

Free

40 M configs

23

Give up?

Example of conventional effective mass plot

N= 46 fermionsL=12

• noise• drift• worse for larger N

Page 18: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Effective mass plot with standard technique

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

meff���E

Free

40 M configs

23

Give up?

Example of conventional effective mass plot

N= 46 fermionsL=12

• noise• drift• worse for larger N

Is there any information here?

Page 19: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

c

Ρ

Τ�32Τ�16Τ�8Τ�4N=4 correlator

distribution at different times (τ)

Look at raw correlator probability distributions:

Long tails at late times

Page 20: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

...but look at distribution for LOG of correlator:

�2 0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

1.2

log�c�

Ρ

Τ�32Τ�16Τ�8Τ�4

Distribution ofLog[correlator]with Gaussian fit

Page 21: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

c

Ρ

Τ�32Τ�16Τ�8Τ�4

�2 0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

1.2

log�c�

Ρ

Τ�32Τ�16Τ�8Τ�4

C ln C

Page 22: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

0 50 100 150 200 250 3000.00

0.01

0.02

0.03

0.04

c

Ρ

Τ�32Τ�16Τ�8Τ�4

�2 0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

1.2

log�c�

Ρ

Τ�32Τ�16Τ�8Τ�4

• Correlators seem to flow toward a log-normal distribution(which is described by only two parameters)

• Noise and drift in measurement due to problems sampling long tail for computing <C>

C ln C

Page 23: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

Page 24: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

Probability distribution: P (x)

Page 25: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x)

Page 26: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

− lnφ(t) = t�x� +t2

2��x2� − �x�2

�+ . . .

=∞�

n=1

tn

n!κn

Cumulant generating function:

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x)

Page 27: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

− lnφ(t) = t�x� +t2

2��x2� − �x�2

�+ . . .

=∞�

n=1

tn

n!κn

Cumulant generating function:

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x)

Analogue withpath integral

Page 28: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

− lnφ(t) = t�x� +t2

2��x2� − �x�2

�+ . . .

=∞�

n=1

tn

n!κn

Cumulant generating function:

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x) Like

e-S

Analogue withpath integral

Page 29: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

Like par

tition

function

Z[J]

− lnφ(t) = t�x� +t2

2��x2� − �x�2

�+ . . .

=∞�

n=1

tn

n!κn

Cumulant generating function:

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x) Like

e-S

Analogue withpath integral

Page 30: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

Like par

tition

function

Z[J]

Like eff

ective

action W

[J]− lnφ(t) = t�x� +

t2

2��x2� − �x�2

�+ . . .

=∞�

n=1

tn

n!κn

Cumulant generating function:

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x) Like

e-S

Analogue withpath integral

Page 31: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Digression: statistics & the RG

Like par

tition

function

Z[J]

Like eff

ective

action W

[J]

nth cumulant - like n-pt. operators in effective action, increasing dimension

− lnφ(t) = t�x� +t2

2��x2� − �x�2

�+ . . .

=∞�

n=1

tn

n!κn

Cumulant generating function:

φ(t) = �e−tx� = 1 − t�x� +t2

2�x2� + . . .

Moment generating function:

Probability distribution: P (x) Like

e-S

Analogue withpath integral

Page 32: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

The Central Limit Theorem as RG flow

x

P(x)P(x) = some probability distribution with zero mean, unit variance.

P (0, 1,κ3,κ4, . . . ;x)Characterize by cumulants:

Page 33: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

The Central Limit Theorem as RG flow

x

P(x)P(x) = some probability distribution with zero mean, unit variance.

P (0, 1,κ3,κ4, . . . ;x)Characterize by cumulants:

Average pairwise: x1 →x1 + x2√

2, x2 →

x3 + x4√2

, . . .(and rescale)

Page 34: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

The Central Limit Theorem as RG flow

x

P(x)P(x) = some probability distribution with zero mean, unit variance.

P (0, 1,κ3,κ4, . . . ;x)Characterize by cumulants:

Cumulants get rescaled:

κn → 2(1−n/2)κn

Average pairwise: x1 →x1 + x2√

2, x2 →

x3 + x4√2

, . . .(and rescale)

Page 35: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

The Central Limit Theorem as RG flow

x

P(x)P(x) = some probability distribution with zero mean, unit variance.

P (0, 1,κ3,κ4, . . . ;x)Characterize by cumulants:

Repeat: P ⇒ P (0, 1, 0, 0, . . . ;x)

... P flows to normal distribution

Cumulants get rescaled:

κn → 2(1−n/2)κn

Average pairwise: x1 →x1 + x2√

2, x2 →

x3 + x4√2

, . . .(and rescale)

Page 36: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

The Central Limit Theorem as RG flow

x

P(x)P(x) = some probability distribution with zero mean, unit variance.

P (0, 1,κ3,κ4, . . . ;x)Characterize by cumulants:

Repeat: P ⇒ P (0, 1, 0, 0, . . . ;x)

... P flows to normal distribution

mean is re

levant

variance

is margi

nal

higher n

: irrelev

antCumulants get rescaled:

κn → 2(1−n/2)κn

Average pairwise: x1 →x1 + x2√

2, x2 →

x3 + x4√2

, . . .(and rescale)

Page 37: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

In our case: correlator C(Φ) driven toward log-normal distribution

log[C(Φ)] driven toward normal distribution

If cumulants κn of log[C(Φ)] behave as irrelevant operators, is there equivalent of effective field theory approach?

Page 38: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

In our case: correlator C(Φ) driven toward log-normal distribution

log[C(Φ)] driven toward normal distribution

If cumulants κn of log[C(Φ)] behave as irrelevant operators, is there equivalent of effective field theory approach?

YES, truncate exact relation:

cumulants of lnC

ln�C� =∞�

n=1

κn

n!

κn are computed from finite sample.

Page 39: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Back to real data for N=46 unitary fermions (40 M configs.)

Effective mass plot with standard technique

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

meff���E

Free

40 M configs

23

Give up?

N= 46 fermionsL=12

Conventional effective mass plotalready shown:

Page 40: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Applying the cumulant expansion for the same data• conventional effective mass plot (gray)• cumulant expansion (blue) to order n ln�C� =

� κn

n!

Page 41: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Cumulant expansion method - Results

��� � � � � �

� �� �

� �� � �

� � �� � � � � �

� � � � � � � �� � � � � � � � � � � � � � �

�����

������

����

� �

��

0 10 20 30 40 500.2

0.3

0.4

0.5

0.6

0.7

0.8

Τ

meff2���E

Free

�� � � � � � � � � � � � � � � � � � � � � � �

� � � � � �� �

�� �

�� � �

��� � �

� � �

� � ��

�����

������

����

� �

��

0 10 20 30 40 500.2

0.3

0.4

0.5

0.6

0.7

0.8

Τ

meff3���E

Free

���� � � � � � � � � �

�� �

����

� �

� ��� �

��

��

� ��

� �

�� �

��

�����

������

����

� �

��

0 10 20 30 40 500.2

0.3

0.4

0.5

0.6

0.7

0.8

Τ

meff4���E

Free

���� � � �

�� �

��

� �

� �

�� �

��

�����

������

����

� �

��

0 10 20 30 40 500.2

0.3

0.4

0.5

0.6

0.7

0.8

Τ

meff5���E

Free

26

n=2

Applying the cumulant expansion for the same data• conventional effective mass plot (gray)• cumulant expansion (blue) to order n ln�C� =

� κn

n!

Page 42: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Cumulant expansion method - Results

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n=3

Cumulant expansion method - Results

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Applying the cumulant expansion for the same data• conventional effective mass plot (gray)• cumulant expansion (blue) to order n ln�C� =

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n!

Page 43: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Cumulant expansion method - Results

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Cumulant expansion method - Results

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n=3

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Applying the cumulant expansion for the same data• conventional effective mass plot (gray)• cumulant expansion (blue) to order n ln�C� =

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Page 44: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Cumulant expansion method - Results

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n=2

Applying the cumulant expansion for the same data• conventional effective mass plot (gray)• cumulant expansion (blue) to order n ln�C� =

� κn

n!

Page 45: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Determination of ground state energy for N=46 from cumulant expansion of log[C]

Cumulant expansion method - Results

2 3 4 50.36

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Page 46: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Determination of ground state energy for N=46 from cumulant expansion of log[C]

Cumulant expansion method - Results

2 3 4 50.36

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Page 47: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Are distributions approaching log-normal appearing in QCD?

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Are distributions approaching log-normal appearing in QCD?

t�5

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6000

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10000

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distribution of log CΛΛEach curve: 100,000 samples

Apparently yes, although time dependence seems to be different

Page 49: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Why do almost log-normal distributions arise?Typically, multiplicative stochastic processes.

Similar physics in electron propagation in random media

Try mean field treatment for probability distribution(inspired by Smolyarenko, Altschuler, 1997)

• Fracturing of materials• Flow of oil through porous rock• ...

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Mean field argument: distribution for y = Log[CN(Φ,T)]

P (y) =�

[Dφ] e−R

d4x 12 m2φ2

δ(lnCN [φ, T ]− y)

N particlecorrelator time separation T

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Mean field argument: distribution for y = Log[CN(Φ,T)]

Perform semiclassical expansion in Φ.

P (y) =�

[Dφ] e−R

d4x 12 m2φ2

δ(lnCN [φ, T ]− y)

N particlecorrelator time separation T

Page 52: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Mean field argument: distribution for y = Log[CN(Φ,T)]

Perform semiclassical expansion in Φ.

Leading order result: • Log Normal distribution (with corrections at higher order); • μ, σ2 scale with N and T as seen in data

P (y) =�

[Dφ] e−R

d4x 12 m2φ2

δ(lnCN [φ, T ]− y)

N particlecorrelator time separation T

Page 53: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

0 10 20 30 40 50 600.8

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1.8

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velocity of variance

LO mean field prediction

LO mean field prediction

Page 54: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Useful to have an analytically soluble toy model: 1 particle, one spatial site

Page 55: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Useful to have an analytically soluble toy model: 1 particle, one spatial site

⇓⇑

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t

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y

(1 + gφ(x, t))

Page 56: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Useful to have an analytically soluble toy model: 1 particle, one spatial site

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(1 + gφ)

(1 + gφ)

Toy model:

C(T ) =T�

i=1

(1 + gφi)

φi ∈ [−1, 1] uniform dist.

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

E(T ) ≡ − 1T

ln�C(T )�φ = 0

Exact answer for the “energy”:

Compare with simulation (finite sample size), g=1/2

Toy model:

C(T ) =T�

i=1

(1 + gφi)

φi ∈ [−1, 1] uniform dist.

Page 58: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

E(T ) ≡ − 1T

ln�C(T )�φ = 0

Exact answer for the “energy”:

Compare with simulation (finite sample size), g=1/2

Toy model:

C(T ) =T�

i=1

(1 + gφi)

φi ∈ [−1, 1] uniform dist.

E → − 1T

ln

�1N

N�

i=1

C(T,φi)

�Two strategies:

•Conventional:

•New “EFT” approach: use identityln�C� =

n

κn

n!κn = cumulants of lnC(T,φ)

estimate κn from sample for low n.

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

50 250 500 1000

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0.01

Κ2Κ3

Eτ = − 1τ lnCτ

Eτ = − 1τ

�12 κ̄2 + 1

6 κ̄3

�Eτ = − 1

τ

�12 κ̄2

τ

nmax = 2nmax = 3

Effective mass plot for toy model

Simulation: sample size N=50,000(each dot)

Eexact: E=0

κ1, κ2, κ3

κ1, κ2

κ1, κ2

conventional

Analytic results:

κ1, κ2, κ3

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Intriguing observation about toy model:improvement if reweighted by mean field solution

(Endres)

0 10 20 30 40 500

2

4

6

8

10

Τ

C�Τ�

sample size = 5000

conventional weighting

reweighted: shifted by mean field solution

Page 61: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Intriguing observation about toy model:improvement if reweighted by mean field solution

(Endres)

0 10 20 30 40 500

2

4

6

8

10

Τ

C�Τ�

sample size = 5000

conventional weighting

reweighted: shifted by mean field solution

Model toosimplistic

...but is there some

way to use m.f.t. to reweight a real

field theory?

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David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

Conclusions:

Page 63: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

•Noise distribution in multi-fermion simulations can have nearly universal structure

Conclusions:

Page 64: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

•Noise distribution in multi-fermion simulations can have nearly universal structure

•Amenable to analysis similar to EFT -- characterized by hierarchy of correlations...can extract information

Conclusions:

Page 65: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

•Noise distribution in multi-fermion simulations can have nearly universal structure

•Amenable to analysis similar to EFT -- characterized by hierarchy of correlations...can extract information

•Possible clues to effective reweighting from mean field expansion of noise distribution

Conclusions:

Page 66: CAPTCHA: Telling Humans and Computers Apart …CAPTCHA: Telling Humans and Computers Apart Automatically A CAPTCHA is a program that protects websites against bots by generating and

David B. Kaplan ~ Lattice 2011 ~ July 15, 2011

•Noise distribution in multi-fermion simulations can have nearly universal structure

•Amenable to analysis similar to EFT -- characterized by hierarchy of correlations...can extract information

•Possible clues to effective reweighting from mean field expansion of noise distribution

•Might similar structure be hiding in other noisy systems (e.g., disconnected diagrams)?

Conclusions: