An introduction to large deviation theory
and some of its applications
Hugo Touchette
School of Mathematical Sciences
Queen Mary, University of London
African Institute for Mathematical Sciences (AIMS)
28 February 2012
Hugo Touchette (QMUL)
Large deviations
February 2012
1 / 27
February 2012
2 / 27
Outline
Themes
Random variables / stochastic systems
Most probable values / typical trajectories
Fluctuations around these values
Small vs large deviations / fluctuations / rare events
1
Examples of large deviations
2
Basic results
3
Applications
Hugo Touchette (QMUL)
Large deviations
Example: Sum of Gaussian random variables
n
1X
Sn =
Xi ,
n
p(Xi = x) = √
i=1
n Sn
1
2πσ 2
2 /(2σ 2 )
e −(x−µ)
Sn
500
4
400
3
300
2
200
1
100
0
0
-1
0
100
200
300
400
500
0
100
200
n
300
400
500
n
Basic observations
Sn → µ in probability
√
Fluctuations ∼ 1/ n → 0
Hugo Touchette (QMUL)
Large deviations
February 2012
4 / 27
Sum of Gaussian random variables (cont’d)
Probability density function (pdf) of Sn :
r
n −n(s−µ)2 /(2σ2 )
p(Sn = s) =
e
2πσ 2
σ2
Variance: var(Sn ) =
→0
n
p(Sn = s)
Dominant part:
n=500
p(Sn = s) ≈ e −nI (s)
I(s)
n=100
Rate function:
(s − µ)2
I (s) =
2σ 2
Hugo Touchette (QMUL)
n=10
µ
Large deviations
s
February 2012
5 / 27
Example: Sum of exponential random variables
n Sn
n
1X
Sn =
Xi
n
Sn
500
3.5
400
3
300
2.5
200
2
i=1
1
p(x) = e −x/µ
µ
1.5
100
1
0
0
100
200
300
400
0
500
100
200
n
300
400
500
n
p(Sn =s)
n=500
Large deviation probability:
p(Sn = s) ≈ e −nI (s)
I(s)
Rate function:
n=100
n=10
s
s
− 1 − ln
µ
µ
I (s) =
Hugo Touchette (QMUL)
µ
Large deviations
s
February 2012
6 / 27
Example: Random bits
Sequence of bits:
ω = |010011100101
{z
},
n bits
P(0) = p0
P(1) = p1 = 1 − p0
Empirical vector:

# 0’s in ω 

Ln,0 (ω) =


n
Ln = (Ln,0 , Ln,1 )

# 1’s in ω 


Ln,1 (ω) =
n
Large deviation probability
P(Ln = µ) ≈ e −nD(µ) ,
D(µ) = µ0 ln
µ0
µ1
+ µ1 ln
p0
p1
D = relative entropy
Zero of D: µ = (p0 , p1 )
Ln → (p0 , p1 ) in probability
Hugo Touchette (QMUL)
Large deviations
February 2012
7 / 27
Example: Spin system
Spin chain (configuration):
ωi ∈ {−1, +1}
ω = ω1 , ω2 , . . . , ωn ,
|
{z
}
n spins
Mean magnetization:
n
1X
Mn =
ωi
n
i=1
Density of states:
Ω(m) = # configurations ω with Mn = m
Large deviation form
Ω(m) ≈ e ns(m) ,
s(m) = −
Hugo Touchette (QMUL)
1−m 1−m 1+m 1+m
ln
−
ln
2
2
2
2
Large deviations
February 2012
8 / 27
Large deviation theory
Random variable: An
Probability density: p(An = a)
Large deviation principle (LDP)
p(An = a) ≈ e −nI (a)
Meaning of ≈:
ln p(a) = −nI (a) + o(n)
1
lim − ln p(a) = I (a)
n→∞ n
Rate function: I (a) ≥ 0
Goals of large deviation theory
1
Prove that a large deviation principle exists
2
Calculate the rate function
Hugo Touchette (QMUL)
Large deviations
February 2012
10 / 27
Varadhan’s Theorem
Exponential expectation:
Z
nf (An )
he
i = p(An = a) e nf (a) da
Assume LDP for An :
p(An = a) ≈ e −nI (a)
Abel Prize 2007
Theorem: Varadhan (1966)
1
lnhe nf (An ) i = max{f (a) − I (a)}
n→∞ n
a
λ(f ) = lim
Special case: f (a) = ka
λ(k) = max{ka − I (a)}
a
λ(k) = Legendre transform of I (a)
Hugo Touchette (QMUL)
Large deviations
February 2012
11 / 27
Gärtner-Ellis Theorem
Scaled cumulant generating function (SCGF)
1
lnhe nkAn i,
n→∞ n
λ(k) = lim
k ∈R
Theorem: Gärtner (1977), Ellis (1984)
If λ(k) is differentiable, then
1
Existence of LDP:
p(An = a) ≈ e −nI (a)
2
Rate function:
Richard S. Ellis
I (a) = max{ka − λ(k)}
k
UMass, USA
I (a) = Legendre transform of λ(k)
I (a) convex in this case
Hugo Touchette (QMUL)
Large deviations
February 2012
12 / 27
Contraction principle
An −→ Bn
LDP
LDP for An :
LDP?
p(An = a) ≈ e −nIA (a)
Contraction: Bn = f (An )
Probability for Bn :
Z
p(Bn = b) =
f −1 (b)
p(An = a) da
Contraction principle
LDP for Bn :
p(Bn = b) ≈ e −nIB (b)
Rate function:
IB (b) =
min IA (a) = min IA (a)
f −1 (b)
a:f (a)=b
Hugo Touchette (QMUL)
Large deviations
February 2012
13 / 27
Sums of IID random variables
Cramér (1938)
Random variable:
n
1X
Xi ,
Sn =
n
i=1
Xi ∼ p(x),
IID
SCGF:
* n
+
Y
1
1
λ(k) = lim lnhe nkSn i = lim ln
e kXi = lnhe kX i
n→∞ n
n→∞ n
i=1
Gaussian
λ(k) = µk +
Poisson
σ2
k 2,
2
(s − µ)2
I (s) =
,
2σ 2
Hugo Touchette (QMUL)
k ∈R
s∈R
λ(k) = − ln(1 − µk),
I (s) =
Large deviations
s
s
− 1 − ln ,
µ
µ
k<
1
µ
s>0
February 2012
14 / 27
Sanov’s Theorem
n IID random variables: ω = ω1 , ω2 , . . . , ωn
Empirical frequencies:
n
Ln,j
1X
# (ωi = j)
=
δωi ,j =
n
n
i=1
Empirical vector: Ln = (Ln,1 , Ln,2 , . . . , Ln,q )
SCGF:
q
X
λ(k) = ln
pj e kj
j=1
Gärtner-Ellis Theorem
LDP: P(Ln = µ) ≈ e −nD(µ)
Rate function: D(µ) = k(µ) · µ − λ(k(µ)) =
Hugo Touchette (QMUL)
Large deviations
q
X
j=1
µj ln
µj
pj
February 2012
15 / 27
Markov processes
Donsker and Varadhan (1975)
Markov chain:
ω = ω1 , ω2 , . . . , ωn ,
p(ω) = ρ(ω1 )π(ω2 |ω1 ) · · · π(ωn |ωn−1 )
Additive process:
n
1X
Sn =
f (ωi )
n
i=1
Gätner-Ellis Theorem
SCGF:
λ(k) = ln ζ(πk )
I
I
Dominant eigenvalue: ζ(πk )
Tilted transition matrix: πk (ωn |ωn−1 ) = π(ωn |ωn−1 )e kf (ωn )
LDP:
p(Sn = s) ≈ e −nI (s) ,
Hugo Touchette (QMUL)
I (s) = max{ks − λ(k)}
Large deviations
k
February 2012
16 / 27
General properties
Most probable value = min and zero of I
Zero of I = Law of Large Numbers
Parabolic minimum = Central Limit Theorem
pn (a)
pn (a)
I(a)
I(a)
a
a
I (a) 6= max{ka − λ(k)} when I is nonconvex
k
Hugo Touchette (QMUL)
Large deviations
February 2012
17 / 27
Noise-perturbed dynamical systems
Dynamical system:
ẋ(t) = F (x(t))
Perturbed dynamics:
Ẋ (t) = F (X (t)) +
√
ξ(t)
| {z }
perturbation
Low-noise limit: → 0
Gaussian white noise:
hξ(t)ξ(t 0 )i = δ(t − t 0 )
hξ(t)i = 0,
x0
x(t)
0
Hugo Touchette (QMUL)
x0
τ
Large deviations
Xǫ (t)
0
τ
February 2012
19 / 27
LDP for the random paths
Functional LDP
p[x] ≈ e
−J[x]/
1
J[x] =
2
,
Z
0
T
[ẋ(t) − F (x(t))]2 dt
{z
}
|
Lagrangian
|
{z
}
Action
Low-noise limit: → 0
Derived in maths by Freidlin and Wentzell (1984)
Derived in physics by Onsager and Machlup (1953)
Zero of J[x] = most probable dynamics = unperturbed dynamics:
ẋ ∗ = F (x ∗ )
Gaussian fluctuations around x ∗ (t)
Hugo Touchette (QMUL)
Large deviations
February 2012
20 / 27
Other LDPs by contraction
Transition probability:
p(x, T |x0 ) ≈ e
V (x, T |x0 ) =
x(t)
x
−V (x,T |x0 )/
min
J[x]
x0
0
x(t):x(0)=x0 ,x(T )=x
Stationary distribution:
x(t)
...
x
p(x) ≈ e −V (x)/
V (x) =
τ
min
J[x]
x(t):x(−∞)=0,x(0)=x
0
...
0
−∞
∂D
Exit time:
τ ≈ e V
∗ /
in probability
V ∗ = min min V (x, t|xs )
x∈∂D t≥0
Hugo Touchette (QMUL)
D
t
Large deviations
February 2012
21 / 27
Example: Ornstein-Uhlenbeck process
System:
√
ẋ(t) = −γx(t) +
ξ(t)
Stationary distribution:
p(x) ≈ e −V (x)/ ,
V (x) =
min
I [x]
x(t):x(0)=x0 ,x(∞)=x
Euler-Lagrange equation:
δI [x ∗ ] = 0
d ∂L ∂L
−
= 0,
dt ∂ ẋ
∂x
⇐⇒
1
L = (ẋ + γx)2
2
Solution: V (x) = I [x ∗ ] = γx 2
General result for gradient systems
ẋ = −∇U(x) +
√
ξ(t)
Hugo Touchette (QMUL)
=⇒
V (x) = 2U(x)
Large deviations
February 2012
22 / 27
Example: Noisy Van der Pol oscillator
ẋ = v
√
v̇ = −x + v (α − x 2 − v 2 ) +
ξ(t)
Bifurcation: Stable fixed point (α < 0) to stable limit cycle (α > 0)
2
2
1
1
v
0
0
�1
�1
0
10
20
30
�2
�2
40
�1
0
t
20
15
10
5
0
�2
r4
2
1
W (r ) = −αr
+
2
0
2
�1
�1
0
1
2
Hugo Touchette (QMUL)
�2
2
x
Stationary distribution: p(r , θ) ≈ e −W (r )/
40
30
20
10
0
�2
1
4
2
1
0
�1
�1
0
1
2
Large deviations
�2
2
2
0
�2
�2
1
0
�1
�1
0
1
2
�2
February 2012
23 / 27
Additive processes
Stochastic process: x(t)
Observable:
1
AT [x] =
T
T
Z
f (x(t)) dt
0
Gärtner-Ellis
SCGF:
1
λ(k) = lim
lnhe TkAT i,
T →∞ T
he
TkAT
i=
Z
D[x] p[x] e TkAT [x]
Rate function: I (a) = max{ka − λ(k)}
k
Donsker-Varadhan
Generator: L
Tilted generator: Lk = L + kf
SCGF: λ(k) = ζ(Lk )
Hugo Touchette (QMUL)
Large deviations
February 2012
24 / 27
Example: Pulled Brownian particle
Langevin dynamics:
mẍ(t) = −α
ẋ −k[x(t) − vt] + ξ(t)
|{z}
|
{z
} |{z}
drag
spring force
noise
Q
Work:
1
WT =
T
Z
vt
U
T
T
F (t) v dt = ∆U + QT
0
Large deviation principle
SCGF:
1
lnhe TkWT i = ck(1 + k),
T →∞ T
λ(k) = lim
Rate function:
Hugo Touchette (QMUL)
c = v2
(w − c)2
I (w ) = max{kw − λ(k)} =
k
4c
Large deviations
February 2012
25 / 27
Other applications
Equilibrium statistical mechanics
I
I
Entropy = rate function
Free energy = SCGF
Multifractals
Chaotic systems (thermodynamic formalism)
Disorded systems
I
I
I
Random walks in random environments
Spin glasses
Quenched/annealed large deviations
Nonequilibrium systems
Interacting particle models
I
I
I
I
I
Zero-range process
Exclusion process
Current, density profile
Fluctuation relations
Space/time large deviations
Simulations of rare events
Hugo Touchette (QMUL)
Large deviations
February 2012
26 / 27
February 2012
27 / 27
Reading
H. Touchette
The large deviation approach to statistical mechanics
Physics Reports 478, 1-69, 2009
arxiv:0804.0327
H. Touchette
A basic introduction to large deviations:
Theory, applications, simulations
arxiv:1106.4146
Y. Oono
Large deviation and statistical physics
Prog. Theoret. Phys. Suppl. 99, 165-205, 1989
A. Dembo and O. Zeitouni
Large Deviations Techniques and Applications
Springer, New York, 1998
Hugo Touchette (QMUL)
Large deviations