Metastability for the Ising model on a random graph

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1 Metastability for the Ising model on a random graph Joint project with Sander Dommers, Frank den Hollander, Francesca Nardi

2 Outline A little about metastability Some key questions A little about the, and the Configuration model random graph Significance of low temperature limit The communication height and some of its properties

3 Concept of metastability We observe (Markovian) dynamics on some state space Ω with energy function H that has a global minimum s and or more local minima m. Dynamics started at m may appear to be in equilibrium, prevented from moving far due to a

4 Concept of metastability We observe (Markovian) dynamics on some state space Ω with energy function H that has a global minimum s and or more local minima m. Dynamics started at m may appear to be in equilibrium, prevented from moving far due to a potential barrier.

5 m and s s is a stable state (trivially unique in our model).

6 m and s s is a stable state (trivially unique in our model). m is a metastable state if no other state has a greater potential barrier between itself and configurations of lower energy (along best path). V (m) V (q)

7 Main questions What can we say about the crossover time In particular, m s?

8 Main questions What can we say about the crossover time m s? In particular, what is the average crossover time E m [τ s ]?

9 Main questions What can we say about the crossover time m s? In particular, what is the average crossover time E m [τ s ]? how does it vary with Ω?

10 Main questions What can we say about the crossover time m s? In particular, what is the average crossover time E m [τ s ]? how does it vary with Ω? what kind of configurations does the dynamics see near the top? We investigate these in the low temperature setting β (this is a parameter of the model).

11 Let G = (V, E) be a multi-graph.

12 Let G = (V, E) be a multi-graph. Configuration (state) space : Ω = {+1, 1} V is the set of +1/ 1 configurations on G.

13 Let G = (V, E) be a multi-graph. Configuration (state) space : Ω = {+1, 1} V is the set of +1/ 1 configurations on G. Hamiltonian on configuration space: for ω Ω, H (ω) = J 2 with J > 0 and h > 0. <u,v> E ω (v) ω (u) h ω (v) 2 v V

14 Let G = (V, E) be a multi-graph. Configuration (state) space : Ω = {+1, 1} V is the set of +1/ 1 configurations on G. Hamiltonian on configuration space: for ω Ω, H (ω) = J 2 with J > 0 and h > 0. Gibbs measure on Ω <u,v> E ω (v) ω (u) h ω (v) 2 µ β (ω) = 1 Z β exp { βh (ω)} v V Assigns exponential preference to configurations with low energy.

15 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ).

16 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ). E.g. for the example below, d = (4, 4, 3, 4, 4, 3)

17 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ). E.g. for the example below, d = (4, 4, 3, 4, 4, 3)

18 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ). E.g. for the example below, d = (4, 4, 3, 4, 4, 3)

19 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ). E.g. for the example below, d = (4, 4, 3, 4, 4, 3)

20 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ). E.g. for the example below, d = (4, 4, 3, 4, 4, 3)

21 Configuration model random graph A family of random graphs that are of particular interest to us, are based on the Configuration model. This begins with a given set of vertices {v 1,..., v n }, and a prescribed degree sequence d = (d 1,..., d n ). E.g. for the example below, d = (4, 4, 3, 4, 4, 3)

22 Glauber Dynamics, significance of β Dynamics on configuration space: continuous-time Glauber dynamics with transition rates {exp ( ) β [H (ω ) H (ω)] c β (ω, ω ) = + if ω ω 0 otherwise where ω ω iff ω and ω differ at exactly one vertex.

23 Glauber Dynamics, significance of β Dynamics on configuration space: continuous-time Glauber dynamics with transition rates {exp ( ) β [H (ω ) H (ω)] c β (ω, ω ) = + if ω ω 0 otherwise where ω ω iff ω and ω differ at exactly one vertex. Note that up moves become exponentially expensive.

24 Glauber Dynamics, significance of β Dynamics on configuration space: continuous-time Glauber dynamics with transition rates {exp ( ) β [H (ω ) H (ω)] c β (ω, ω ) = + if ω ω 0 otherwise where ω ω iff ω and ω differ at exactly one vertex. Note that up moves become exponentially expensive. Bounds on R eff (effective resistance) show that w.h.p. dynamics avoid unnecessarily expensive configurations when going from m s, and will take an optimal path (see for example Metastability - A. Bovier, F. den Hollander).

25 Communication Height The communication height between m and s on G is Γ := Φ (m, s) H (m) = min max γ:m s σ γ H (σ) H (m) - represents the lowest barrier between m and s which the dynamics must overcome. - paths γ that correspond to the minimum are called optimal paths.

26 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

27 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

28 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

29 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

30 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

31 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

32 Communication Height For example, for the previous graph we have the following optimal path for suitable values of J and h (note: easy to show here that m = and s = )

33 Significance of Γ How is Γ related to the crossover time m s? It follows from standard universal metastability theorems that lim exp β ( βγ ) E [τ ] = K Furthermore, in T 2 Z 2 the pre-factor K is related to C, where C := Configurations on optimal paths where height Γ is first seen. We expect that a similar relation holds for our model.

34 Some properties of Γ When min i d i 3, the resulting Configuration model is an expander graph. Therefore Γ = Θ (n), with n = V (under some assumptions on the distribution of d).

35 Some properties of Γ When min i d i 3, the resulting Configuration model is an expander graph. Therefore Γ = Θ (n), with n = V (under some assumptions on the distribution of d). Upper bound on Γ : by a coupling argument, we can show that Γ l n /4 where l n is the total degree of the graph. We would like to make this sharper.

36 Some properties of Γ We also expect that Γ n n C for some constant C depending on the parameters of the model, when d is regular, and possibly also for d i i.i.d. under some constraints.

37 Some properties of Γ We also expect that Γ n n C for some constant C depending on the parameters of the model, when d is regular, and possibly also for d i i.i.d. under some constraints. To this avail, we can show that if A V is a random set/configuration of total degree l A, then ( A L2 l A 1 l ) A 1 d d 1

38 Some properties of Γ A d ( L2 l A 1 l ) A 1 d 1 We need to show that this holds not only for random sets, but also for extremal.

39 Some properties of Γ A d ( L2 l A 1 l ) A 1 d 1 We need to show that this holds not only for random sets, but also for extremal.

40 back to m and s But what are m and s? It is easy to see that s =, i.e. the configuration with +1 assigned at every vertex. It is also easy to see that is a local minimum (for non-degenerate values of J and h). Showing that m = and nothing else (in the case when degree sequence satisfies d i 3) is difficult.

41 Conclusion There are many unforeseen challenges when assigning the Ising model to a random graph. Global properties of the random graph model play a key role. Precise descriptions of Γ, C exist in lattice setting, but may be too difficult for random graphs. But good estimates should be attainable.

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