My PhD
Tom Rafferty
Paul Chleboun
Stefan Grosskinsky
University of Warwick
t.rafferty@warwick.ac.uk
February 24, 2014
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
1 / 19
Overview
1
Introduction
2
Mini-project
3
More recent work
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
2 / 19
Introduction
Key points
Some background from mini-project
Coupling and Attractivity
Particle systems with stationary product measures
Calculate mixing and relaxation times
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
3 / 19
Coupling, Monotonicity and Attractivity
A coupling of two probability distributions µ and ν is a pair of
random variables (X , Y ) defined on a single probability space such
that the marginal distribution of X is µ and the marginal distribution
of Y is ν. That is, a coupling (X , Y ) satisfies P(X = x) = ν(x) and
P(Y = y ) = ν(y )
The partial ordering of two configurations η and ζ defined on a
state-space of the form S = X Λ where Λ is a lattice or network and
for interacting particle systems X ⊆ N. η ≤ ζ if we have ηx ≤ ζx for
all x ∈ Λ
For probability measures µ1 , µ2 on S: µ1 ≤ µ2 provided that
µ1 (f ) ≤ µ2 (f ) for all increasing f
A process (η(t) : t ≥ 0) on S is attractive if the property of
stochastic monotonicity, or the partial ordering of configurations, is
preserved through time.
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
4 / 19
Mini-project
Key Points
Consider the zero-range process (ZRP). S = XL,N
P
Lf (η) = x,z∈Λ u(ηx )p(x, z)(f (η x→z ) − f (η))
Attractive ⇔ u(x) ≤ u(x + 1) for all x ∈ N
Coupling Construction
Consider the joint state space (XL,N , XL,N+1 ) between process η and ξ
such that, ξ = η + δy for some y ∈ ΛL .
ξy = n + 1 o
ηy = n
ξ =n+1 o
y
ηy = n
u(ξy )−u(ηy )
−−−−−−−→
u(ηy )
−−−−−−−−→
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
n ξ =n
y
ηy = n
n ξ =n
y
ηy = n − 1
February 24, 2014
(1)
5 / 19
Some pictures
uHΞ2L-uHΗ2L º uHΗ2+1L-uHΗ2L
1
2
3
4
5
uHΗ2L
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
6 / 19
Stationary measure (Canonical)
Some equations
νφx [ηx = n] =
wx (n) =
wx (n)(φ)n
zx (φ)
n
Y 1
k=1
ux (k)
h X
i IX (η) Y
(η) = N = L,N
πL,N [η] = νφΛ η
wx (ηx ),
ZL,N
L
x∈ΛL
P
Q
where ZL,N = η∈XL,N x∈NΛL wx (ηx )
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
(2)
7 / 19
The coupled generator
Coupled generator
Lf (η, y ) =
X
[u(ηx )p(x, z)(f (η x→z , y ) − f (η, y ))]
x,z∈ΛL
+
X
(u(ηy + 1) − u(ηy ))p(y , z)(f (η, z) − f (η, y )).
(3)
z∈ΛL
Stationary coupled measure
Let µ(η, y ) = µ(y |η)πL,N (η) be the stationary measure of the coupled
process.
µ(Lf (η, y )) = 0 for all f
µ(y |η) = αη (y ) is a possible growth rule
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
8 / 19
Another picture
pN HΗ,1L
1
pN HΗ,4L
2
3
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
4
5
February 24, 2014
9 / 19
A result
A result
If µ(η, y ) is stationary then αη (y ) must satisfy
X
X
uz (ηz )p(x, z)αηz→x (y )+
[uz (ηz + 1) − uz (ηz )] p(z, y )αη (z)
x,z∈ΛL
=
X
x,z∈ΛL
z∈ΛL
ux (ηx )p(x, z)αη (y )+
X
[uy (ηy + 1) − uy (ηy )] p(y , z)αη (y ).
z∈ΛL
(4)
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
10 / 19
Results
0
10
1
Canonical Measure:
π (η ≥ n)
L,N
0.8
⟨ Tailη(n) ⟩η
⟨ Tailη(n)⟩ η
⟨ Tailη(n)⟩ η
−2
10
−4
10
⟨ t1(n) ⟩η
0.6
0.4
0.2
−6
10
0
5
10
0
0
n
1
2
3
4
5
n
Figure: N = 256
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
11 / 19
Results
0
10
1
L,N
0.8
−1
⟨ Tailη(n)⟩ η
10
⟨ Tailη(n)⟩ η
Canonical Measure:
π (η ≥ n)
−2
10
⟨ Tailη(n) ⟩η
⟨ t1(n) ⟩η
0.6
0.4
−3
10
0.2
−4
10
0
5
10
0
0
n
1
2
3
4
5
n
Figure: N = 512
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
12 / 19
Results
0
10
1
L,N
0.8
−2
⟨ Tailη(n)⟩ η
10
⟨ Tailη(n)⟩ η
Canonical Measure:
π (η ≥ n)
−4
10
⟨ Tailη(n) ⟩η
⟨ t1(n) ⟩η
0.6
0.4
−6
10
0.2
−8
10
0
5
10
0
0
n
1
2
3
4
5
n
Figure: N = 512
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
13 / 19
Results
0
10
1
L,N
0.8
−2
⟨ Tailη(n)⟩ η
10
⟨ Tailη(n)⟩ η
Canonical Measure:
π (η ≥ n)
−4
10
⟨ Tailη(n) ⟩η
⟨ t1(n) ⟩η
0.6
0.4
−6
10
0.2
−8
10
0
5
10
0
0
n
1
2
3
4
5
n
Figure: N = 512
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
14 / 19
Mixing and relaxation times
Mixing
d(t) = maxx∈S ||Pt (x, .) − π||TV
-Mixing: tmix () := min{t : d(t) ≤ }
tmix := tmix (1/4)
Relaxation
Smallest non-zero eigenvalue (Reversible chains)
D(f )
λ = inf{ var
: varπ (f ) 6= 0}
π (f )
P
D(f ) = x,y ∈S rx,y |f (y ) − f (x)|2
||Pt f − π(f )||22 ≤ e −2λt Varπ (f )
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
15 / 19
A birth death process
Β2
0
1
Α2
2
3
4
5
Properties
S = {0, 1 . . . , N}
Birth rates αk = 1
Death rates βk = g (k) = 1 +
Stationary distribution π(n) ∝
b
e 1−γ
1−γ
−b −b n1−γ
e
≥
1
g (n)!
b
b
kγ
≈ e b/k
γ
1
g (n)!
≥ e 1−γ e
−b
(n+1)1−γ
1−γ
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
16 / 19
Mixing times are hitting times of large sets (Peres, Sousi
(2013))
Hitting times
τi = inf{t > 0 : Xt = i}
Ti (j) = E(τi |X0 = j)
LTi (j) = −1
Ti (i) = 0
P
PN−s
Ts N = N−s
n=0
k=n
π[N−n]
g (N−k)π[N−k]
Mixing and hitting large sets
Define Aα such that π(Aα ) ≥ α
tH (α) = maxx,Aα :π(Aα ≥α) TAα (x)
There exist
cα , cα0
> 0 such that
(= TAα (N))
0
cα tH (α) ≤ tmix
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
≤ cα tH (α)
February 24, 2014
17 / 19
Results
tmix ∼ N 1+γ
trel ∼ N 2γ
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
18 / 19
The End
Thanks
Thanks
Tom Rafferty Paul Chleboun Stefan Grosskinsky (Complexity
ZRP and
Science)
Condensation
February 24, 2014
19 / 19