ScalingLimits of InteractingParticleSystemspi.math.cornell.edu/files/Probability/Cornell-Lec2.pdf’: B k(G n) !B k(G) withmax v2B k (Gn)d X(x n v;x ’) < ,where B k(G) (respy,B k(G

Scaling Limitsof

Interacting Particle Systems

Kavita RamananDivision of Applied Math

Brown University

11th Cornell Probability Summer SchoolCornell, IthacaJun 17–20, 2019

K. Ramanan Scaling Limits

Scaling Limits of Interacting Particle Systems

Lecture 2Convergence of Interacting Diffusions on Sparse

Graph Sequences


Recap: Networks of locally interacting stochastic processes

We want to study locally interacting stochastic processes on agiven finite connected graph G = (V ,E ):

that evolve as a discrete-time Markov chain of the form:

XG ,vt+1 = F

(XG ,vt , µvt , ξ

vt+1

), v ∈ V ,

where F : S × P(S)× S ′ 7→ S,or as a diffusion of the form

dXG ,vt = b(XG ,v

t , µvt )dt + dW vt , v ∈ V ,

with b : Rd × P(Rd) 7→ Rd ,where in both cases µvt is the local empirical measure at v :

µvt =1dv

∑u∼v

δX ut, v ∈ V .


Key Questions

Sequence of sparse (possibly random) graphs Gn = (Vn,En) with|Vn| → ∞ with bounded (average) degree

XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn,

or

dXGn,vt = b(XGn,v

t , µvt )dt + dW vt , v ∈ Vn,

Specific Questions:(1) Does the whole system admit a scaling limit?(2) Is there an autonomous description of the limiting dynamics ofa typical particle?(3) Does the (global) empirical measure

µ̄Gn =1|Vn|

∑v∈Vn

δXGn,v (·)

have a scaling limit?K. Ramanan Scaling Limits

Networks of interacting stochastic processes

We will focus ondiscrete-time Markov chains and diffusions

and consider the simplest setting to better illustrate the main ideas

but one can consider different interacting stochastic evolutions,including

continuous time jump processes (A. Ganguly),

The research described in these lectures is based on collaborationswith

D. Lacker (Columbia U.), Ruoyu Wu (U. Michigan), A. Ganguly(Brown)

M. Wortsman, T. Sudijono




















Lecture 2 focuses on the convergence question


XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = b(XGn,v

t , µvt )dt + dW vt , v ∈ Vn

(Q1) Does the whole system admit a scaling limit?

(A1) Yes, wrt a small generalization of local weak convergenceIn fact, for this, we can allow more general heterogeneous,time-dependent, dynamics: e.g.,

XGn,vt+1 = F t

v

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = bv (t,XGn,v



Lecture 2 focuses on the convergence question


XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = b(XGn,v


(Q1) Does the whole system admit a scaling limit?(A1) Yes, wrt a small generalization of local weak convergenceIn fact, for this, we can allow more general heterogeneous,time-dependent, dynamics: e.g.,

XGn,vt+1 = F t

v

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = bv (t,XGn,v



Outline of Lecture 2

Part A: Question of Convergence

Background on local weak convergence of graphsExamples of local weak convergence of graphsBackground on local weak convergence of marked graphsStatement of convergence theoremProof of convergence theorem

Part B: Question of Characterizing Marginals


1. Background on local weak convergence of graphs

History and References for local weak convergence

Introduced by Benjamini and Schramm (2001)Further developed by Aldous-Steele in“The Objective Method: Probabilistic CombinatorialOptimization and Local Weak Convergence”Also, Aldous-Lyons “Processes on Unimodular RandomNetworks”, EJP (2007)Charles Bordenave lecture notes:Also (especially for the marked case) see the Lacker-R-Wupaper “Large sparse networks of interacting diffusions’paper(especially the Appendix)https://arxiv.org/abs/1904.02585



◦ A connected rooted graph G = (V ,E , ) is a graph (V ,E )(assumed as usual to be locally finite with either finite or countablevertex set) with a distinguished vertex ρ ∈ V .

Definition: Isomorphisms of rooted graphs

We say two connected rooted graphs Gi = (Vi ,Ei , ρi ) areisomorphic if there exists a bijection ϕ : V1 → V2 such thatϕ(ρ1) = ρ2 and (ϕ(u), ϕ(v)) ∈ E2 if and only if (u, v) ∈ E1, foreach u, v ∈ V1. We denote this by G1 ∼= G2. We refer to the mapϕ as an isomorphism from G1 to G2.

Let I (G ,G ′) denote the set of isomorphisms between two graphsG ,G ′ ∈ G∗

Let G∗ denote the set of isomorphism classes of connected rootedgraphs.



◦ Given k ∈ N and G = (V ,E , ρ) ∈ G∗, let Bk(G ) denote theinduced subgraph of G consisting of those vertices whose graphdistance from ρ is no more than k .

Definition of local weak convergence

We say that a sequence (Gn ∈ G∗)n∈N converges locally to G ∈ G∗if∀k ∃nk s.t. Bk(G ) ∼= Bk(Gn) for all n ≥ nk ,

where ∼= denotes isomorphism.

◦ There is a metric compatible with this notion of convergence thatrenders G∗ a complete and separable space

◦ Thus we can talk of random G∗-valued (rooted) graphs (or rathertheir isomorphism classes) and (usual) weak convergence ofGn ∈ G∗ to a (possibly random) limit G ∈ G∗.



◦ Given k ∈ N and G = (V ,E , ρ) ∈ G∗, let Bk(G ) denote theinduced subgraph of G consisting of those vertices whose graphdistance from ρ is no more than k .

Definition of local weak convergence

We say that a sequence (Gn ∈ G∗)n∈N converges locally to G ∈ G∗if∀k ∃nk s.t. Bk(G ) ∼= Bk(Gn) for all n ≥ nk ,

where ∼= denotes isomorphism.

◦ There is a metric compatible with this notion of convergence thatrenders G∗ a complete and separable space◦ Thus we can talk of random G∗-valued (rooted) graphs (or rathertheir isomorphism classes) and (usual) weak convergence ofGn ∈ G∗ to a (possibly random) limit G ∈ G∗.


2. Examples of local weak convergence of graphs

1. Cycle graph converges to infinite line

ρ −→

...

ρ

...


Examples of local weak convergence

2. Line graph converges to infinite line

ρ −→

...

ρ

...



3. Line graph rooted at end converges to semi-infinite line

ρ

−→

...

ρ



4. Finite to infinite d-regular trees

(A graph is d-regular if ever vertex has degree d .)

ρ −→ρ



5. Uniformly random regular graph to infinite regular tree

Fix d . Among all d-regulargraphs on n vertices, se-lect one uniformly at ran-dom. Place the root ata (uniformly) random ver-tex. When n → ∞,this converges (in law) tothe infinite d-regular tree.(McKay ’81)

−→ρ


The random rooted graph U(G )

Given a finite (possibly disconnected graph) G we write U(G ) forthe random connected rooted graph obtained by assigning a rootuniformly at random and then isolating the connected componentcontaining the root.



6. Erdős-Rényi to Galton-Watson

If Gn = G (n, pn) with npn → p ∈ (0,∞), then both Gn (with aroot assigned uniformly at random) and U(Gn) converge in law toGW(Poisson (p)), the Galton-Watson tree with offspringdistribution Poisson(p).

ρ −→

root

ρ

......


Examples of Local weak convergence

7. Configuration Model

Let Gn be drawn from the configuration model with (empirical)degree distribution converging to some distribution π on N0, withfinite nonzero first moment. Then U(Gn) converges locally in lawto the augmented or unimodular Galton-Watson tree with offspringdistribution π, denoted UGW (π).


Unimodular Galton Watson Tree

The unimodular Galton-Watson tree UGW (π). is a random treedefined so that the root has offspring distribution π and each of itschildren has offspring distribution π̂, where

π̂(k) =(k + 1)π(k + 1)∑

n∈N nπ(n), k ∈ N0.

root

ρ



8. Preferential Attachment Graphs to a Random Tree

A result by Berger-Borgs-Chayes-Saberi (’14) shows convergence ofpreferential attachment graphs to a random tree

Key PointLocal limits of many classes of random graphs are often trees

For references for the different limit theorems, seeLacker-R-Wu paper “Large sparse networks of interacting diffusions”

https://arxiv.org/abs/1904.02585

Of course, deterministic graphs could have local limits of a differentnature

9. Convergence of Finite Lattices

Zκ ∩ [−n, n]κ converges to Zκ



8. Preferential Attachment Graphs to a Random Tree

A result by Berger-Borgs-Chayes-Saberi (’14) shows convergence ofpreferential attachment graphs to a random tree

Key PointLocal limits of many classes of random graphs are often trees

For references for the different limit theorems, seeLacker-R-Wu paper “Large sparse networks of interacting diffusions”

https://arxiv.org/abs/1904.02585

Of course, deterministic graphs could have local limits of a differentnature

9. Convergence of Finite Lattices

Zκ ∩ [−n, n]κ converges to Zκ


Convergence of Interacting Processes on Sparse Graphs

But we want to understand convergence of dynamics on graphs,i.e., as |Vn| → ∞, understand behavior of

XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = b(XGn,v


• We can think of the trajectory XGn,v = {X n,vt , t ∈ T} for T = N0

or T = [0,∞), as a (random) mark associated to the vertex v• the mark lies in X , where X is either SN0 or C := C([0,∞) : Rd)• For any graph G , let XG := {XG ,v , v ∈ V }

Leads us to consider convergence of marked graphs (Gn,XGn)




XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = b(XGn,v



or T = [0,∞), as a (random) mark associated to the vertex v

• the mark lies in X , where X is either SN0 or C := C([0,∞) : Rd)• For any graph G , let XG := {XG ,v , v ∈ V }





XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = b(XGn,v








XGn,vt+1 = F

(XGn,vt , µvt , ξ

vt+1

), v ∈ Vn

or

dXGn,vt = b(XGn,v






Local weak convergence for marked graphs

Let (X , dX ) be a metric space, then a marked (rooted connected)graph is a pair (G , x) such that G = (V ,E , ρ) is rooted connectedgraph and a vector of marks x = (xv )v∈V ∈ XV .

Definition: Isomorphims of marked Graphs

We say that two marked graphs (G , x) and (G ′, x ′) are isomorphicif there exists an isomorphism ϕ from G to G ′ such that(xv )v∈V = (x ′ϕ(v))v∈V . We write (G , x) ∼= (G ′, x ′) to indicateisomorphism. We let G∗[X ] refer to the set of isomorphism classesof marked graphs.


Local weak convergence for marked graphs

Let (X , dX ) be a metric space, then a marked (rooted connected)graph is a pair (G , x) such that G = (V ,E , ρ) is rooted connectedgraph and a vector of marks x = (xv )v∈V ∈ XV .

Definition: Isomorphims of marked Graphs

We say that two marked graphs (G , x) and (G ′, x ′) are isomorphicif there exists an isomorphism ϕ from G to G ′ such that(xv )v∈V = (x ′ϕ(v))v∈V . We write (G , x) ∼= (G ′, x ′) to indicateisomorphism. We let G∗[X ] refer to the set of isomorphism classesof marked graphs.


Local convergence of marked graphsDefinition: Local convergence of marked graphs

We say that a sequence (Gn, xn) ∈ G∗[X ] converges locally to(G , x) ∈ G∗[X ] if, for every k ∈ N and ε > 0, there exist nk ∈ Nsuch that for all n ≥ nk , there exists an isomorphismϕ : Bk(Gn)→ Bk(G ) with maxv∈Bk (Gn) dX (xn

v , xϕ(v)) < ε, whereBk(G ) (respy, Bk(Gn)) is the set of vertices in G (respy, Gn) thatare at most distance k from the root ρ.

The space G∗[X ] can be equipped with the metric

d∗((G , x), (G ′, x ′)) =∞∑k=1

12k

(1 ∧ inf

ϕ∈I (Bk (G),Bk (G ′))max

v∈Bk (G)dX (xv , x

′ϕ(v))

),

where recall I (G ,G ′) denotes the set of isomorphisms between twographs G ,G ′ ∈ G∗.

Exercise: Show that if (X , dX ) is complete and separable then sois (G∗[X ], d∗).






d∗((G , x), (G ′, x ′)) =∞∑k=1

12k

(1 ∧ inf



′ϕ(v))

),








d∗((G , x), (G ′, x ′)) =∞∑k=1

12k

(1 ∧ inf



′ϕ(v))

),




A property of local convergence of marked graphs

Lemma

Suppose Gn → G in G∗. Suppose, for each n, XGn = (X v ,Gn)v∈Gn

is a XGn -valued random variable. Suppose finally that for each k ,the family

{X v ,Gn : n ∈ N, v ∈ Bk(Gn)}

of X -valued random variables is tight. Then (Gn,XGn) is tight inG∗[X ].


Convergence Results: Discrete-time Markov Chains

• Given a marked graph (G , x) ∈ G∗[S], consider the chain

XG ,vt+1 = Ft

(XG ,vt , µG ,vt , ξG ,vt+1

), v ∈ V , t ∈ N0,

with initial condition XG0 = x . Denote soln. with init cond. as XG ,x

Assumptions (Lacker-R-Wu ’19)

We assume the following conditions are satisfied:1 For each (possibly disconnected) graph G , {ξvt : v ∈ G , t ∈ N}

are iid S ′-valued with common law θ ∈ P(S).2 Ft : S × P(S)× S ′ 7→ S is continuous for each t ∈ N0

• Let PG ,x = Law(XG ,x) ∈ P((S∞)V )



Now consider the sequence XGn = {XGn,v , v ∈ Gn} where

XGn,vt+1 = Ft

(XGn,vt , µGn,v

t , ξGn,vt+1

), v ∈ Vn, t ∈ N,

with initial condition XGn0 = xn,

Theorem (Lacker-R-Wu ’18)

Under the stated assumptions1 the map (G , x) 7→ PG ,x is continuous2 as n→∞, if (Gn, xn)→ (G , x) in G∗[S], then

(Gn,XGn , xn)⇒ (G ,XG ,x).In particular, the trajectory of the root particle XGn,ρ converges inlaw in S∞ to XG ,ρ, the trajectory of the root particle of thelimiting graph.



XG ,x is the unique solution XG = {XG ,v , v ∈ V }, where

XG ,vt+1 = Ft


), v ∈ V , t ∈ N0,

with initial condition XG0 = x ∈ SV .

Note (G , x) ∈ G∗[S]

and (G ,XG ,x) ∈ G∗[S∞]

Basic Idea of ProofShowing that the map (G , x) 7→ (G ,XG ,x) from G∗[S] to G∗[S∞]is continuous is equivalent to showing that for every t ∈ N0 and

bounded and continuous map ϕ on G∗[Sk+1], the function

ψ(G , x) = E[ϕ(G ,XG ,xt )].

is continuous on G∗[S].




XG ,vt+1 = Ft


), v ∈ V , t ∈ N0,


Note (G , x) ∈ G∗[S] and (G ,XG ,x) ∈

G∗[S∞]



ψ(G , x) = E[ϕ(G ,XG ,xt )].





XG ,vt+1 = Ft


), v ∈ V , t ∈ N0,


Note (G , x) ∈ G∗[S] and (G ,XG ,x) ∈ G∗[S∞]



ψ(G , x) = E[ϕ(G ,XG ,xt )].





XG ,vt+1 = Ft


), v ∈ V , t ∈ N0,


Note (G , x) ∈ G∗[S] and (G ,XG ,x) ∈ G∗[S∞]



ψ(G , x) = E[ϕ(G ,XG ,xt )].



Outline of Proof of Convergence for Markov Chains

Need to show that for every t ∈ N0 and bounded and continuousmap ϕ on G∗[St+1], the function

ψ(G , x) = E[ϕ(G ,XG ,xt )]


Fix r ∈ N. It suffices to verify that the following function iscontinuous:

ψr (G , x) := ϕ(Br (G ),XBr (G),xBr (G)

t ).

Key realization in the simpler Markov chain setting: on anyinterval [0, t], the behavior of particles in a Br -neighborhoodof the root only depend on the initial conditions in aBr+t-neighborhood of the root. This dependence is continuousdue to the continuity of each Fs , s ≤ t.The conclusion then follows from the fact that(Gn, xn)→ (G , x) in G∗[X ] implies that the projections of theinitial conditions converge: xn

ϕ(Br+t(Gn)) → xBr+t(G)




ψ(G , x) = E[ϕ(G ,XG ,xt )]

is continuous on G∗[S].Fix r ∈ N. It suffices to verify that the following function iscontinuous:


t ).






ψ(G , x) = E[ϕ(G ,XG ,xt )]



t ).

Key realization in the simpler Markov chain setting: on anyinterval [0, t], the behavior of particles in a Br -neighborhoodof the root only depend on the initial conditions in aBr+t-neighborhood of the root. This dependence is continuousdue to the continuity of each Fs , s ≤ t.

The conclusion then follows from the fact that(Gn, xn)→ (G , x) in G∗[X ] implies that the projections of theinitial conditions converge: xn





ψ(G , x) = E[ϕ(G ,XG ,xt )]



t ).




Wasserstein Metric

Definition

Let P1(Rd) be the subspace of measures µ ∈ P(Rd) that satisfy∫X|y |µ(dy) <∞.

Then a sequence of measures µn ∈ P1(Rd) is said to converge toµ ∈ P1(Rd) in the Wasserstein metric if µn converges to µ weaklyand the first moments also converge:∫

Rd

|y |µn(dy)→∫Rd

|y |µ(dy).

Note: P1(Rd), equipped with the Wasserstein-1 metric, is Polish.


Convergence Results: Diffusion Setting

XG = (XG ,v )v∈V , where

dXGn,vt = b(XGn,v

t , µGn,vt )dt + dW v

t , v ∈ V ,

with initial condition XGn0 = xn, and

µGn,vt =

1dv

∑u∼v

δXGn,ut

, v ∈ V .

Assumptions in the Diffusion SettingSuppose the following properties hold:

1 The initial conditions are iid with common distribution θ thathas a finite second moment.

2 The drift function b : Rd × P1(Rd) 7→ Rd is Lipschitzcontinuous and has linear growth:

b(x ,m) ≤ C (1 + |x |+∫|y |m(dy).


Convergence Results: Interacting Diffusions

XG = (XG ,v )v∈V , where

dXGn,vt = b(XGn,v


t , v ∈ V ,

with initial condition XGn0 = xn, where b : Rd ×P1(Rd) 7→ Rd , and

µGn,vt =

1dv

∑u∼v

δXGn,ut

, v ∈ V .

Theorem (Lacker-R-Wu ’19)

Under the stated assumptions, there is a unique strong solution tothe SDE and and as n→∞, if (Gn, xn)→ (G , x) in G∗[S], then(Gn,XGn,xn

)⇒ (G ,XG ,x).


Diffusion Convergence Results: Outline of ProofXG = (XG ,v )v∈V , where

dXGn,vt = b(XGn,v


t , v ∈ V , XGn0 = x .

1 Use stochastic calculus to show that under the assumption, forevery T <∞,

supG∈G∗

supv∈G

E

[sup

t∈[0,T ]|XG ,v

t |2]<∞.

2 Apply Aldous’ tightness criterion for measures in P(C), whereC is the space of Rd -valued continuous functions on [0,∞), toshow that the family of C-valued random variables{XG ,v ,G ∈ G∗, v ∈ G} is tight.

3 Use above lemma on local convergence of marked graphs toconclude that (Gn,X

Gn) is tight as a sequence of G∗[C]random elements.

4 Use stochastic calculus to show that the limit is a (weak)solution of the limit SDE (with Gn replaced with G ), andinvoke uniqueness of the latter.



dXGn,vt = b(XGn,v


t , v ∈ V , XGn0 = x .


supG∈G∗

supv∈G

E

[sup

t∈[0,T ]|XG ,v

t |2]<∞.







dXGn,vt = b(XGn,v


t , v ∈ V , XGn0 = x .


supG∈G∗

supv∈G

E

[sup

t∈[0,T ]|XG ,v

t |2]<∞.







dXGn,vt = b(XGn,v


t , v ∈ V , XGn0 = x .


supG∈G∗

supv∈G

E

[sup

t∈[0,T ]|XG ,v

t |2]<∞.




4 Use stochastic calculus to show that the limit is a (weak)solution of the limit SDE (with Gn replaced with G ), andinvoke uniqueness of the latter.K. Ramanan Scaling Limits

Part B of Lecture 2: Local Characterization of Dynamics

We have shown that XGn,v can be approximated by a dynamicalsystem on an infinite graph G :

XG ,vt+1 = F

(XG ,vt , µvt , ξ

vt+1

), v ∈ V ,

where F : S × P(S)× S ′ 7→ S,or as a diffusion of the form

dXG ,vt = b(XG ,v


with b : Rd × P(Rd) 7→ Rd ,where in both cases µvt is the local empirical measure at v :

µvt =1dv

∑u∼v

δX ut, v ∈ V .

Specific Questions:(2) Is there an autonomous description of the limiting dynamics ofa typical particle?


A Well-Studied Static Analog: Markov Random Fields

Notation: For a set A of vertices in a graph G = (V ,E ), define

Boundary: ∂A = {u ∈ V \A : ∃u ∈ A s.t. u ∼ v},

State space S; {Y v , v ∈ V } canonical variables acting on SVDefn. A probability measure π on SV is said to be a MarkovRandom Field if for π a.e. ηA,

π(Y A = ηA|Y V \A = ηV \A

)= π

(Y A = ηA|Y ∂A = η∂A

)

A

A

δ


Markov Random Fields on Trees

An Equivalent Formulation: (Y v )v∈V is a Markov random fieldwrt G = (V ,E ) if for finite A ⊂ V , B ⊂ V \ [A ∪ ∂A],

(Y v )v∈A ⊥ (Y v )v∈B | (Y v )v∈∂A,

Xr

X1 X2 X3

X11 X12 X21 X22 X31 X32

• Tree structure allows one to analyze the marginal distribution ofMRF at a node.


Markov Random Fields

Markov Random Fields or Gibbs measures on Trees are much easierto analyze on trees

e.g. Ising model on the tree

(Yv )v∈A ⊥ (Yv )v∈B | (Yv )v∈∂A,

Xr

X1 X2 X3

X11 X12 X21 X22 X31 X32

Use the implied MRF conditional independence structure to createa recursion, and analyze that recursion to compute the marginal at

the rootK. Ramanan Scaling Limits

In Search of a Conditional Independence Property

XG ,vt+1 = F

(XG ,vt , µvt , ξ

vt+1

), v ∈ V ,

dXG ,vt = b(XG ,v


Question A:Is the MRF property retained: that is, if (X v

0 )v∈V are iid, at eachtime t will (X v

t )v∈V form a Markov random field?

Answer ANo!



XG ,vt+1 = F

(XG ,vt , µvt , ξ

vt+1

), v ∈ V ,

dXG ,vt = b(XG ,v


Question A:Is the MRF property retained: that is, if (X v

0 )v∈V are iid, at eachtime t will (X v

t )v∈V form a Markov random field?

Answer ANo!



dXG ,vt = b(XG ,v


Question B:Does the following space-time Markov-random field property hold:For each time t, if (X v

0 )v∈V are iid, do the particle trajectories(X v [t])v∈V form a Markov random field?

Here, x [t] = (x(s), s ∈ [0, t]).



dXG ,vt = b(XG ,v


Question B:Does the following space-time Markov-random field property hold:For each time t, if (X v

0 )v∈V are iid, do the particle trajectories(X v [t])v∈V form a Markov random field?

Here, x [t] = (x(s), s ∈ [0, t]).



dXG ,vt = b(XG ,v


A

∂A

Reformulation of Question B:Given t > 0, if (X v

0 )v∈V are iid, for any finite A ⊂ V , is

XA[t] ⊥ XV\[A∪∂A][t]|X∂A[t]?

Answer BNo!



dXG ,vt = b(XG ,v


A

∂A

Reformulation of Question B:Given t > 0, if (X v

0 )v∈V are iid, for any finite A ⊂ V , is

XA[t] ⊥ XV\[A∪∂A][t]|X∂A[t]?

Answer BNo!


Second-order Markov Random Fields

Double Boundary∂2A = ∂A ∪ [∂(∂A) \ A]

A

∂A

∂(∂A)\A

A

Double Boundary

Definition: A family of random variables (Y v )v∈V is a 2nd-orderMarkov random field if

(Y v )v∈A ⊥ (Y v )v∈B | (Y v )v∈∂2A,

for all finite sets A,B ⊂ V with B ∩ (A ∪ ∂2A) = ∅.



dXG ,vt = b(XG ,v


Question C:If (X v

0 )v∈V are iid, at each time t, do the states (X v (t))v∈V form asecond-order Markov random field?

(X vt )v∈A ⊥ (X v

t )v∈B | (X vt )v∈∂2A,


Answer C:NO



dXG ,vt = b(XG ,v


Question C:If (X v

0 )v∈V are iid, at each time t, do the states (X v (t))v∈V form asecond-order Markov random field?

(X vt )v∈A ⊥ (X v

t )v∈B | (X vt )v∈∂2A,


Answer C:NO


At Last A Conditional Independence Property

dX v (t) =1dv

∑u∼v

b(X vt ,X

ut )dt + dW v

t , (X v0 )v∈V i.i.d.

Question D:If (X v

0 )v∈V are iid, at each time t, do the particle trajectories(X v [t])v∈V form a second-order Markov random field?

Theorem 4: (Lacker, R, Wu ’17) YES!

(X v [t])v∈A ⊥ (X v [t])v∈B | (X v [t])v∈∂2A,

for all A ⊂ V finite and B ⊂ V with B ∩ (A ∪ ∂2A) = ∅.

In fact, suffices for (X v0 )v∈V to form a second-order MRF.



dX v (t) =1dv

∑u∼v

b(X vt ,X

ut )dt + dW v

t , (X v0 )v∈V i.i.d.

Question D:If (X v








dX v (t) =1dv

∑u∼v

b(X vt ,X

ut )dt + dW v

t , (X v0 )v∈V i.i.d.

Question D:If (X v







Comments on the Conditional Independence Property

Relevant to work on Gibbs-non-Gibbs transitions – mostlystudied when G = Zm (e.g., Dereudre-Roelly ’05;Redig-Roelly-Ruszel ’10)

We also need an extension to the case when A is infinite:

Theorem (Lacker, R, Wu ’18)


for all A ⊂ V , B ⊂ V with B ∩ (A ∪ ∂2A) = ∅.



Comments on the Conditional Independence Property

Relevant to work on Gibbs-non-Gibbs transitions – mostlystudied when G = Zm (e.g., Dereudre-Roelly ’05;Redig-Roelly-Ruszel ’10)We also need an extension to the case when A is infinite:

Theorem (Lacker, R, Wu ’18)


for all A ⊂ V , B ⊂ V with B ∩ (A ∪ ∂2A) = ∅.



Documents

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