AIMS 2014 — Session on Nonlocally Coupled Dynamical Systems
A group-theoretic approach to metastability
in networks of interacting SDEs
Nils Berglund
MAPMO, Université d’Orléans
Madrid, July 8, 2014
With Sébastien Dutercq (Orléans),
Bastien Fernandez (Marseille/Paris) and Barbara Gentz (Bielefeld)
Nils Berglund
nils.berglund@univ-orleans.fr
http://www.univ-orleans.fr/mapmo/membres/berglund/
Interacting SDEs with noise
Example 1 [B, Fernandez, Gentz, Nonlinearity 2007]
.
N particles on a circle Z /NZ
.
Bistable local dynamics
.
Ferromagnetic nearest neighbour coupling
.
Independent noise on each site
x7
x0
x6
x5
x1
x2
x4
x3
√
γ
dxti = xti − (xti )3 dt + xti+1 − 2xti + xti−1 dt + 2ε dWti
2
√
dxt = −∇V (xt ) dt + 2ε dWt
X
γ X i+1
potential V (x) =
U(x i ) +
(x
− x i )2
U(ξ) = 14 ξ 4 − 12 ξ 2
4
Gradient system
i
i
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
1 / 14
Interacting SDEs with noise
Example 1 [B, Fernandez, Gentz, Nonlinearity 2007]
.
N particles on a circle Z /NZ
.
Bistable local dynamics
.
Ferromagnetic nearest neighbour coupling
.
Independent noise on each site
x7
x0
x6
x5
x1
x2
x4
x3
√
γ
dxti = xti − (xti )3 dt + xti+1 − 2xti + xti−1 dt + 2ε dWti
2
√
dxt = −∇V (xt ) dt + 2ε dWt
X
γ X i+1
potential V (x) =
U(x i ) +
(x
− x i )2
U(ξ) = 14 ξ 4 − 12 ξ 2
4
Gradient system
i
i
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
1 / 14
Interacting SDEs with noise
Example 1 [B, Fernandez, Gentz, Nonlinearity 2007]
.
N particles on a circle Z /NZ
.
Bistable local dynamics
.
Ferromagnetic nearest neighbour coupling
.
Independent noise on each site
x7
x0
x1
x6
x5
x2
x3
x4
√
γ
dxti = xti − (xti )3 dt + xti+1 − 2xti + xti−1 dt + 2ε dWti
2
√
dxt = −∇V (xt ) dt + 2ε dWt
X
γ X i+1
potential V (x) =
U(x i ) +
(x
− x i )2
U(ξ) = 14 ξ 4 − 12 ξ 2
4
Gradient system
i
i
Example 2 [B, Dutercq, preprint 2013]: Same potential + constraint
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
P
i
xi = 0
1 / 14
General gradient systems with noise
dxt = −∇V (xt ) dt +
√
2ε dWt
V : RN → R : confining potential, class C 2
.
Stationary points: X = {x : ∇V (x) = 0}
.
Local minima: X0 = {x ∈ X : all ev of Hessian ∇2 V (x) are > 0}
.
Saddles of index 1: X1 = {x ∈ X : ∇2 V (x) has 1 negative ev }
Dynamics ∼ markovian jump process on G = (X0 , E), E ⊂ X1
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
2 / 14
General gradient systems with noise
dxt = −∇V (xt ) dt +
√
2ε dWt
V : RN → R : confining potential, class C 2
.
Stationary points: X = {x : ∇V (x) = 0}
.
Local minima: X0 = {x ∈ X : all ev of Hessian ∇2 V (x) are > 0}
.
Saddles of index 1: X1 = {x ∈ X : ∇2 V (x) has 1 negative ev }
Dynamics ∼ markovian jump process on G = (X0 , E), E ⊂ X1
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
2 / 14
General gradient systems with noise
dxt = −∇V (xt ) dt +
√
2ε dWt
V : RN → R : confining potential, class C 2
.
Stationary points: X = {x : ∇V (x) = 0}
.
Local minima: X0 = {x ∈ X : all ev of Hessian ∇2 V (x) are > 0}
.
Saddles of index 1: X1 = {x ∈ X : ∇2 V (x) has 1 negative ev }
Dynamics ∼ markovian jump process on G = (X0 , E), E ⊂ X1
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
2 / 14
Eyring–Kramers law
Definition: Communication height
H(xi∗ , xj∗ ) =
inf
γ:xi∗ →xj∗
sup V (γt ) − V (xi∗ )
∗
z2∗ = z21
t
∗
z3∗ = z32
= V (zij∗ ) − V (xi∗ )
H3 = H(x3∗ , x2∗ )
H2 = H(x2∗ , x1∗ )
Definition: Metastable hierarchy
x1∗ ≺ x2∗ ≺ · · · ≺ xn∗ ⇔ ∃θ > 0: ∀k
∗ })
Hk := H(xk∗ , {x1∗ , . . . , xk−1
∗
∗
∗
∗
, . . . , xk∗ }) − θ
6 min H(xi , {x1 , . . . , xi−1 , xi+1
i<k
A group-theoretic approach to metastability in networks of interacting SDEs
x3∗
x2∗
x1∗
July 8, 2014
3 / 14
Eyring–Kramers law
Definition: Communication height
H(xi∗ , xj∗ ) =
inf
γ:xi∗ →xj∗
sup V (γt ) − V (xi∗ )
∗
z2∗ = z21
t
∗
z3∗ = z32
= V (zij∗ ) − V (xi∗ )
H3 = H(x3∗ , x2∗ )
H2 = H(x2∗ , x1∗ )
Definition: Metastable hierarchy
x1∗ ≺ x2∗ ≺ · · · ≺ xn∗ ⇔ ∃θ > 0: ∀k
∗ })
Hk := H(xk∗ , {x1∗ , . . . , xk−1
∗
∗
∗
∗
, . . . , xk∗ }) − θ
6 min H(xi , {x1 , . . . , xi−1 , xi+1
i<k
Theorem: Eyring–Kramers law,
x3∗
x2∗
x1∗
[Bovier,Eckhoff,Gayrard,Klein 2004]
λk = k th ev of generator
τk = first-hitting times
of nbh of {x1∗ , . . . , xk∗ }
|det ∇2 V (zk∗ )| Hk /ε 2π
∗
E xk [τk−1 ] =
e
1 + Oε (1) ' λ−1
k
∗
∗
2
|λ− (zk )| det ∇ V (xk )
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
3 / 14
Potential landscape for Example 1
V (x) =
X
i∈Z /NZ
U(x i ) +
γ
4
X
(x i+1 − x i )2
U(ξ) = 14 ξ 4 − 12 ξ 2
i∈Z /NZ
γ = 0: X = {−1, 0, 1}N , X0 = {−1, 1}N , X1 = {x ∈ X : one x i = 0}
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
4 / 14
Potential landscape for Example 1
V (x) =
X
U(x i ) +
i∈Z /NZ
γ
4
X
(x i+1 − x i )2
U(ξ) = 14 ξ 4 − 12 ξ 2
i∈Z /NZ
γ = 0: X = {−1, 0, 1}N , X0 = {−1, 1}N , X1 = {x ∈ X : one x i = 0}
Theorem
[BFG, Nonlinearity 2007]
No bifurcation for 0 6 γ 6
where γ ∗ (N) >
1
4
γ ∗ (N)
∀N > 2
Vγ (zγ∗ ) = V0 (z0∗ ) + γ(# interfaces) + · · ·
Ising-like dynamics
A group-theoretic approach to metastability in networks of interacting SDEs
+++++
++
−+++++
−++++−
−+++−−
−−++−−
−−+−−−
−−−−−−
July 8, 2014
4 / 14
Potential landscape for Example 1
V (x) =
X
U(x i ) +
i∈Z /NZ
γ
4
X
(x i+1 − x i )2
U(ξ) = 14 ξ 4 − 12 ξ 2
i∈Z /NZ
γ = 0: X = {−1, 0, 1}N , X0 = {−1, 1}N , X1 = {x ∈ X : one x i = 0}
Theorem
[BFG, Nonlinearity 2007]
No bifurcation for 0 6 γ 6
where γ ∗ (N) >
1
4
γ ∗ (N)
∀N > 2
Vγ (zγ∗ ) = V0 (z0∗ ) + γ(# interfaces) + · · ·
Ising-like dynamics
Theorem
γ>
+++++
++
−+++++
−++++−
−+++−−
−−++−−
−−+−−−
−−−−−−
[BFG, Nonlinearity 2007]
1
2 sin2 (π/N)
⇔ X0 = {±(1, . . . , 1)}, X1 = {0} ⇔ Synchronization
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
4 / 14
Transition to synchronization
Symmetry group
G = DN × Z 2 = hr , s, ci
r (x) = (x 2 , x 3 , . . . , x N , x 1 )
s(x) = (x N , x N−1 , . . . , x 1 )
c(x) = −x
X partitionned into
group orbits Ox = {gx : x ∈ G }
Stabilizer: Gx = {g ∈ G : gx = x}
⇒ |Ox ||Gx | = |G |
Useful to study bifurcation diagram
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
5 / 14
Transition to synchronization
Symmetry group
G = DN × Z 2 = hr , s, ci
2
3
N
N−1
0
N
1
r (x) = (x , x , . . . , x , x )
s(x) = (x , x
1
,...,x )
c(x) = −x
X partitionned into
group orbits Ox = {gx : x ∈ G }
Stabilizer: Gx = {g ∈ G : gx = x}
Useful to study bifurcation diagram
γ̃1
[×2]
(1, 1, 1, 1)
[×1]
(0, 0, 0, 0)
[×2]
(1, −1, 1, −1)
2
5
1
2
2
3
1
γ̃2
γ
(x, x, x, x)
(0, 0, 0, 0)
[×4]
(1, 0, 1, 0)
[×8]
(1, 0, 1, −1)
[×8]
(1, −1, 0, 0)
[×8]
(0, 1, 0, 0)
[×4]
(1, 0, −1, 0)
[×4]
(1, 1, −1, −1)
[×8]
(1, 1, 0, 0)
[×16]
(1, 1, 0, −1)
[×8]
(1, 1, 1, −1)
[×8]
(1, 1, 1, 0)
(x, −x, x, −x)
I±
O
A(2)
(x, y, x, y)
(x, y, x, z)
(x, −x, y, −y)
(x, y, x, z)
B
(x, 0, −x, 0)
A
(x, x, −x, −x)
(x, x, y, y)
Aa
(x, y, z, t)
(x, y, x, z)
(x, y, x, z)
I+
Aaα
⇒ |Ox ||Gx | = |G |
1
3
γ!
∂a
I − ∂b
Aaα
∂a
∂b
Aaα
A
I+
Aa
I+
A
I+
A
I−
I−
I+
A
I−
O
I−
Figure 1
Example: N = 4, |X0 | = 81
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
5 / 14
Transition to synchronization
Symmetry group
G = DN × Z 2 = hr , s, ci
2
3
N
N−1
0
N
1
r (x) = (x , x , . . . , x , x )
s(x) = (x , x
1
,...,x )
c(x) = −x
X partitionned into
group orbits Ox = {gx : x ∈ G }
Stabilizer: Gx = {g ∈ G : gx = x}
Useful to study bifurcation diagram
γ̃1
[×2]
(1, 1, 1, 1)
[×1]
(0, 0, 0, 0)
[×2]
(1, −1, 1, −1)
2
5
1
2
2
3
1
γ̃2
γ
(x, x, x, x)
(0, 0, 0, 0)
[×4]
(1, 0, 1, 0)
[×8]
(1, 0, 1, −1)
[×8]
(1, −1, 0, 0)
[×8]
(0, 1, 0, 0)
[×4]
(1, 0, −1, 0)
[×4]
(1, 1, −1, −1)
[×8]
(1, 1, 0, 0)
[×16]
(1, 1, 0, −1)
[×8]
(1, 1, 1, −1)
[×8]
(1, 1, 1, 0)
(x, −x, x, −x)
I±
O
A(2)
(x, y, x, y)
(x, y, x, z)
(x, −x, y, −y)
(x, y, x, z)
B
(x, 0, −x, 0)
A
(x, x, −x, −x)
(x, x, y, y)
Aa
(x, y, z, t)
(x, y, x, z)
(x, y, x, z)
I+
Aaα
⇒ |Ox ||Gx | = |G |
1
3
γ!
∂a
I − ∂b
Aaα
∂a
∂b
Aaα
A
I+
Aa
I+
A
I+
A
I−
I−
I+
A
O
I−
I−
Figure 1
Example: N = 4, |X0 | = 81
Problem: no metastable hierarchy ⇒ Usual Eyring–Kramers law invalid
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
5 / 14
Representation theory of finite groups
.
Representation: π : G 7→ GL(n, C ) s.t. π(gh) = π(g )π(h) ∀g , h ∈ G
.
π, π 0 are equivalent ⇔ ∃S ∈ GL(n, C ) : Sπ(g )S −1 = π 0 (g ) ∀g ∈ G
π (g ) 0
π reducible if equiv to π 0 , π 0 (g ) = 10 π2 (g ) i.e. π = π1 ⊕ π2
.
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
6 / 14
Representation theory of finite groups
.
Representation: π : G 7→ GL(n, C ) s.t. π(gh) = π(g )π(h) ∀g , h ∈ G
.
π, π 0 are equivalent ⇔ ∃S ∈ GL(n, C ) : Sπ(g )S −1 = π 0 (g ) ∀g ∈ G
π (g ) 0
π reducible if equiv to π 0 , π 0 (g ) = 10 π2 (g ) i.e. π = π1 ⊕ π2
.
Theorem
.
A finite group G has finitely many irreducible repr. π (0) , . . . , π (r −1)
.
Any repr. π admits a unique (up to order) decomposition
L −1 (p) (p)
P (p)
π = rp=0
α π
dim(π (p) ) = dim(π)
pα
.
Projector on invariant subspace associated with π (p)
(p) ) P
(p)
P (p) = dim(π
χ(p) (g ) = Tr(π (p) (g ))
g ∈G χ (g )π(g )
|G |
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
6 / 14
Representation theory of finite groups
.
Representation: π : G 7→ GL(n, C ) s.t. π(gh) = π(g )π(h) ∀g , h ∈ G
.
π, π 0 are equivalent ⇔ ∃S ∈ GL(n, C ) : Sπ(g )S −1 = π 0 (g ) ∀g ∈ G
π (g ) 0
π reducible if equiv to π 0 , π 0 (g ) = 10 π2 (g ) i.e. π = π1 ⊕ π2
.
Theorem
.
A finite group G has finitely many irreducible repr. π (0) , . . . , π (r −1)
.
Any repr. π admits a unique (up to order) decomposition
L −1 (p) (p)
P (p)
π = rp=0
α π
dim(π (p) ) = dim(π)
pα
.
Projector on invariant subspace associated with π (p)
(p) ) P
(p)
χ(p) (g ) = Tr(π (p) (g ))
P (p) = dim(π
g ∈G χ (g )π(g )
|G |
Example: Dihedral group DN = hr , si, r N = s 2 = id, rs = sr −1
N even: 4 1-dim irred repr: π(r i s j ) = (±1)i (±1)j ,
A group-theoretic approach to metastability in networks of interacting SDEs
N
2
− 1 2-dim irred repr
July 8, 2014
6 / 14
Markovian jump processes with symmetries
Generator L: transition rates Lij =
cij
mi
e−hij /ε ,
cij = cji
∀i, j ∈ X0
Assumptions
.
Reversibility: mi e−Vi /ε Lij = mj e−Vj /ε Lji
.
Symmetry: π(g )L = Lπ(g ) ∀g ∈ G
where π(g )ab = 1{g (a)=b} permutation matrix
.
Metastable order on the set of G -orbits A1 ≺ · · · ≺ Am
.
No accidental degeneracy:
ha1 b1 = ha2 b2 ⇔ ∃g ∈ G : g ({a1 , b1 }) = {a2 , b2 }
A group-theoretic approach to metastability in networks of interacting SDEs
∀i, j ∈ X0
July 8, 2014
7 / 14
Markovian jump processes with symmetries
Generator L: transition rates Lij =
cij
mi
e−hij /ε ,
cij = cji
∀i, j ∈ X0
Assumptions
.
Reversibility: mi e−Vi /ε Lij = mj e−Vj /ε Lji
.
Symmetry: π(g )L = Lπ(g ) ∀g ∈ G
where π(g )ab = 1{g (a)=b} permutation matrix
.
Metastable order on the set of G -orbits A1 ≺ · · · ≺ Am
.
No accidental degeneracy:
ha1 b1 = ha2 b2 ⇔ ∃g ∈ G : g ({a1 , b1 }) = {a2 , b2 }
∀i, j ∈ X0
Main observation
P (p) L = LP (p)
p = 0, . . . , r − 1
⇒ each subspace P (p) C n is invariant for L
The generator restricted to this subspace satisfies an asymmetric
Eyring–Kramers law
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
7 / 14
Modified Eyring–Kramers law
Method: find basis of P (p) C n , compute matrix elements, apply asym algo
Trivial representation: π (0) (g ) = 1 ∀g
⇒
m ev (m = # orbits)
Theorem [B, Dutercq 2013]
τk = first-hitting time of A1 ∪ A2 ∪ · · · ∪ Ak , µ uniform measure on Ak
E µ [τk−1 ] =
1 + O(e−θ/ε )
|Gai ∩ Gaj | mak Hk /ε e
1 + O(e−θ/ε ) =
(0)
|Gak | cai aj
λ
k
where ak ∈ Ak , (ai , aj ) relevant saddle for Hk := H(Ak , A1 ∪ · · · ∪ Ak−1 )
Other 1-dim representations:
Ai is active for π (p) ⇔ π (p) (h) = 1 ∀h ∈ Ga , a ∈ Ai
Similar result for process on set of active orbits
Representations of dim > 2:
Explicit expression for L(p) = L|P (p) C n (involving characters χ(p) = Tr π (p) )
(p)
(p)
(p) (p)
(p)
Generically, ev are those of blocks Lii or Lii − Lij (Ljj )−1 Lji
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
8 / 14
Modified Eyring–Kramers law
Method: find basis of P (p) C n , compute matrix elements, apply asym algo
Trivial representation: π (0) (g ) = 1 ∀g
⇒
m ev (m = # orbits)
Theorem [B, Dutercq 2013]
τk = first-hitting time of A1 ∪ A2 ∪ · · · ∪ Ak , µ uniform measure on Ak
E µ [τk−1 ] =
1 + O(e−θ/ε )
|Gai ∩ Gaj | mak Hk /ε e
1 + O(e−θ/ε ) =
(0)
|Gak | cai aj
λ
k
where ak ∈ Ak , (ai , aj ) relevant saddle for Hk := H(Ak , A1 ∪ · · · ∪ Ak−1 )
Other 1-dim representations:
Ai is active for π (p) ⇔ π (p) (h) = 1 ∀h ∈ Ga , a ∈ Ai
Similar result for process on set of active orbits, inactive orbits =
Representations of dim > 2:
Explicit expression for L(p) = L|P (p) C n (involving characters χ(p) = Tr π (p) )
(p)
(p)
(p) (p)
(p)
Generically, ev are those of blocks Lii or Lii − Lij (Ljj )−1 Lji
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
8 / 14
Modified Eyring–Kramers law
Method: find basis of P (p) C n , compute matrix elements, apply asym algo
Trivial representation: π (0) (g ) = 1 ∀g
⇒
m ev (m = # orbits)
Theorem [B, Dutercq 2013]
τk = first-hitting time of A1 ∪ A2 ∪ · · · ∪ Ak , µ uniform measure on Ak
E µ [τk−1 ] =
1 + O(e−θ/ε )
|Gai ∩ Gaj | mak Hk /ε e
1 + O(e−θ/ε ) =
(0)
|Gak | cai aj
λ
k
where ak ∈ Ak , (ai , aj ) relevant saddle for Hk := H(Ak , A1 ∪ · · · ∪ Ak−1 )
Other 1-dim representations:
Ai is active for π (p) ⇔ π (p) (h) = 1 ∀h ∈ Ga , a ∈ Ai
Similar result for process on set of active orbits, inactive orbits =
Representations of dim > 2:
Explicit expression for L(p) = L|P (p) C n (involving characters χ(p) = Tr π (p) )
(p)
(p)
(p) (p)
(p)
Generically, ev are those of blocks Lii or Lii − Lij (Ljj )−1 Lji
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
8 / 14
Example 2 for N = 4
V (x) =
P
i
U(x i ) +
γ
4
P
i (x
i+1
− x i )2 restricted to {
For 0 6 γ < 2/5: |X0 | = 6
A1 = orbit of (1, 1, −1, −1) + O(γ)
A2 = orbit of (1, −1, 1, −1) + O(γ)
π = 2π+++ ⊕ π−++ ⊕ π−−− ⊕ π1,−
παβγ (r i s j c k ) = αi β j γ k
i
π1,− (r i s j c k ) = 0i −0 i
01 j
10
(−1)k
A group-theoretic approach to metastability in networks of interacting SDEs
−
−
+
+
−
+
−
+
P
+
−
−
+
−
+
+
−
x i = 0}
+
−
+
−
July 8, 2014
+
+
−
−
9 / 14
Example 2 for N = 4
V (x) =
P
i
U(x i ) +
γ
4
P
i (x
i+1
− x i )2 restricted to {
For 0 6 γ < 2/5: |X0 | = 6
A1 = orbit of (1, 1, −1, −1) + O(γ)
A2 = orbit of (1, −1, 1, −1) + O(γ)
π = 2π+++ ⊕ π−++ ⊕ π−−− ⊕ π1,−
παβγ (r i s j c k ) = αi β j γ k
i
π1,− (r i s j c k ) = 0i −0 i
01 j
10
(−1)k
A group-theoretic approach to metastability in networks of interacting SDEs
−
−
+
+
−
+
−
+
P
+
−
−
+
−
+
+
−
x i = 0}
+
−
+
−
July 8, 2014
+
+
−
−
9 / 14
Example 2 for N = 4
V (x) =
P
i
U(x i ) +
γ
4
P
i (x
i+1
− x i )2 restricted to {
For 0 6 γ < 2/5: |X0 | = 6
A1 = orbit of (1, 1, −1, −1) + O(γ)
A2 = orbit of (1, −1, 1, −1) + O(γ)
π = 2π+++ ⊕ π−++ ⊕ π−−− ⊕ π1,−
παβγ (r i s j c k ) = αi β j γ k
i
π1,− (r i s j c k ) = 0i −0 i
La1 a2 =
La2 a1 =
La1 a10 =
ca1 a2
ma1
ca1 a2
ma2
ca1 a1
ma1
01 j
10
e−ha2 a1 /ε
−ha
e
0 /ε
1 a1
La1 a2 La1 a10 La2 a1
−
+
+
(−1)k
A1
e−ha1 a2 /ε
−
A2
dim
−
+
−
+
P
+
−
−
+
−
+
+
−
x i = 0}
+
−
+
−
+
−
−
eigenvalues
π+++
1
1
2
0, −4La2 a1
π−++
1
0
1
−4La1 a10
π−−−
0
1
1
−4La2 a1
π1,−
2
0
2
(−2La1 a10 )×2
|A|
4
2
6
6
A group-theoretic approach to metastability in networks of interacting SDEs
+
July 8, 2014
graph
+
+
A1
− −
+ +
A1
− −
−
+
A2
−
+
−
+
A2
−
+
+
+
A1
−
−
9 / 14
Example 2 for N = 8
For small γ: |X0 | = 182, |X1 | = 560
32
−++
− A1 +
− −+
32
32
β αα
β A12 β
ββα
32
32
−++
+ A5 −
− −+
β αα
β A9 α
βββ
32
16
32
−++
− A3 +
+− −
β αα
β A8 β
βαβ
64
16
32
−++
− A4 +
−+−
βαβ
β A10 α
αβ β
32
32
−++
− A2 −
++−
βαβ
β A11 α
βαβ
16
−+−
+ A7 +
−+−
32
32
−++
− A6 −
+−+
32
A group-theoretic approach to metastability in networks of interacting SDEs
32
+ = 1, − = −1
√
√
α = 5/ 19, β = −3/ 19
July 8, 2014
10 / 14
Example 2 for N = 8
# eigenvalues: Tr(P (p) |Ai ) =
dim π (p)
|Ga |
X
χ(p) (h)
∀a ∈ Ai
h∈Ga
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
dim
π+++
π++−
π+−+
π+−−
π−++
π−+−
π−−+
π−−−
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
1
0
0
0
0
1
1
1
0
0
1
1
0
0
1
1
0
0
0
1
0
0
0
1
0
0
1
1
0
0
1
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
12
6
2
4
7
2
6
9
π1,+
π1,−
π2,+
π2,−
π3,+
π3,−
0
2
2
0
0
2
0
0
0
2
0
0
2
2
2
2
2
2
2
2
2
2
2
2
0
2
2
0
0
2
2
2
2
2
2
2
0
0
0
0
0
0
4
4
4
4
4
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
4
4
4
4
20
24
24
22
20
24
|A|
8
4
16
16
8
16
2
32
16
16
16
32
182
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
11 / 14
Example 2 for N = 8
Irreducible representation π−++
−++
− A1 +
− −+
−++
− A4 +
−+−
β αα
β A12 β
ββα
β αα
β A8 β
βαβ
−++
+ A5 −
− −+
−++
− A6 −
+−+
A group-theoretic approach to metastability in networks of interacting SDEs
−++
− A2 −
++−
July 8, 2014
12 / 14
Example 2 for N = 8
Irreducible representations of dimension 2: graph of successors
−++
− A1 +
− −+
β αα
β A9 α
βββ
−++
− A4 +
−+−
β αα
β A12 β
ββα
−++
− A3 +
+− −
β αα
β A8 β
βαβ
−++
+ A5 −
− −+
βαβ
β A10 α
αβ β
−++
− A6 −
+−+
(p)
(p)
βαβ
β A11 α
βαβ
−++
− A2 −
++−
(p)
(p)
(p)
−+−
+ A7 +
−+−
(p)
Eigenvalues are those of L22 , . . . , L12,12 and L11 − L19 (L99 )−1 L91
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
13 / 14
Concluding remarks
.
Also understood: behaviour at bifurcations, det ∇2 (z ∗ ) = 0
.
Also understood: Example 1 in limit N → ∞: Allen–Cahn SPDE
.
In progress: from Markovian jump processes to diffusions
.
In progress: Example 2 for arbitrary N
References
. N. B., Bastien Fernandez and Barbara Gentz, Metastability in interacting nonlinear
stochastic differential equations I: From weak coupling to synchronisation,
Nonlinearity 20, 2551–2581 (2007)
II: Large-N behaviour, Nonlinearity 20, 2583–2614 (2007)
. N. B. and Barbara Gentz, The Eyring–Kramers law for potentials with
nonquadratic saddles, Markov Processes Relat. Fields 16, 549–598 (2010)
. N. B. and Barbara Gentz, Sharp estimates for metastable lifetimes in parabolic
SPDEs: Kramers’ law and beyond, Electronic J. Probability 18, (24):1–58 (2013)
. N. B. and Sébastien Dutercq, The Eyring–Kramers law for Markovian jump
processes with symmetries, preprint arXiv:1312.0835 (2013)
A group-theoretic approach to metastability in networks of interacting SDEs
July 8, 2014
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