Ising models on random graphs
Prioriello Maria Luisa
A.A 2011/2012
PhD school in Multiscale Modelling, Computational Simulations and
Characterization in Material and Life Sciences
1
Introduction
During the rst year of the PhD school, we dealt with the study of Ising spin models on lattices
and on random graphs. In particular:
(1a) We studied the classical Curie-Weiss (CW) model on a lattice. The Curie-Weiss model
is an exactly solvable mean-eld model which is a prototype for the ferromagnetic phase
transition, [3] and [4].
Despite that CW be a very well known model, no results on its nite size corrections seem
to be available in the literature. Therefore, the rst part of our activity has been devoted
to this issue using both numerical and analytical methods. In particular we started by
comparing the critical temperature in the thermodynamic limit with the apparent critical
temperature in nite size model. The numerical analysis has been performed using two
dierent algorithms.
(1b) Through a theoretical computation we calculated explicitly the rst correction term for
the principal observables of the model and aslo the second correction term for the magnetization. Then we compared the theoretical computation with the numerical data
obtained with the simulations.
(2) In [1] and [2] is considered a ferromagnetic Ising model on random graphs with power-law
degree distribution. In these papers the model is solved in the sense that the existence
of the pressure in the thermodynamic limit is proven and its explicit expression is given
as a function of a family of random elds (called cavity elds) satisfying a distributional
xed point equation. The same elds determine also magnetization and internal energy
of the model.
In collaboration also with Remco Van Der Hofstad (Eindhoven University of Technology),
visiting professor of the PhD school in October 2012, we started the study of the Law of
Large Numbers (LLN) and the Central Limit Theorem (CLT) for the total spin SN of
the graph of volume N. The rst step has been to identify the measures with respect to
which the two theorems can be formulated. In fact here a double source of stocasticity is
present: the randomness of the spin congurations and the randomness of the graph. The
interplay between the two has to be taken into account in describing the thermodynamic
properties of the model.
1
2
Methods
(1a) Algorithm based on Monte Carlo method (MC) implemented in Fortran. The Monte
Carlo method aims at a probabilistic description, relying on the use of random numbers.
In fact MC method generates stochastic trajectory that visit the phase space with frequencies that reproduce the probability of the Boltzmann-Gibbs distribution.
On the other hand, the mean-eld CW hamiltonian can be expressed as a function of the
magnetization and this fact makes it possible also to compute numerically the probability
distribution of the magnetization in a virtually exact way.
The data obtained with the two algorithms were processed in matlab to get other observables of interest and their graphs.
(1b) The nite volume corrections of CW models are computed using asymptotics of integrals,
in particular Laplace method.
(2) Methods of statistical mechanics and probability are used in the study of the limit theorems on the random graph.
3
Results and Discussion
The nite volume corrections can be heuristically studied by inquiring into the approach of
crtitical phenomena (phase transition) as the volume of the CW model is incresed.
(1a) Thus, for all nite size N we can approximate the critical temperature through the Binder
Cumulant intersection method or through the susceptibility; both methods provide an
apparent critical temperature.
The susceptibility is expected to show a singularity in the thermodynamic limit at the
critical tempereature which is known to be Tc = 1. In a nite lattice the susceptibility
cannot diverge, but reaches a maximum of nite height only; the magnitude of this
maximum depends on the size of the lattice (Figure 1) likewise its positions, which
privides the apparent critical temperature.
Figure 1: Susceptibility for dierent sizes
For a Curie-Weiss model with zero external eld, the Binder Cumulant is dened through
the moments of the magnetization per particle mN as:
2
UN = 1 −
4
mN
3 hm2N i
2.
(1)
This quantity is very useful for detecting the existence of phase transitions and, in particular, for obtaining estimates of critical temperature Tc . One may plot UN versus the
themperature T for varius sizes N and estimate Tc from the common intersection point of
these curves: in fact, UN is expected to be indipendent of N at the critical temperature.
However, due to corrections to nite size scaling, there may be some scatter in the intersection points for dierent pairs of curves if one works with very small linear dimension
(Figure 2).
Figure 2: Binder Cumulant
(1b) The explicit computation of the value of pressure of the CW model in the thermodynamic
limit can be done using the Hubbard-Stratonovich identity.
The rst order correction term to the thermodynamic limit can be obtained using the
Laplace metod and a Taylor expansion. In this way, we obtain:
1
ln
pN (β, h) = f (s ) +
N
∗
r
1
1
1
βN
+ ln 2 +
ln 2π −
ln (−f 00 (s∗ )) −
ln N
2π
2N
2N
2N
(2)
where β is the inverse temperature and h the external eld.
The correction of the nite volume magnetization MN (β, h) with respect to the innite
volume magnetization M (β, h) can be obtained by writing MN (β, h) = M (β, h) + aN b .
The result is:
MN (β, h) =
1 ∂
1 β sinh (βh + βs∗ ) cosh (βh + βs∗ )
pN (β, h) = M (β, h) −
2
β ∂h
N
β − cosh2 (βh + βs∗ )
Hence,
b = 1,
a(β) =
β sinh (βh + βs∗ ) cosh (βh + βs∗ )
,
2
β − cosh2 (βh + βs∗ )
(3)
i.e. at the rst order the correction is O(1/N ).
In order to check the analytical computation, we have compared the coecient a(β), given
in (3), with the results of the numerical simulations that produced MN (β, h). By tting
3
the numerical graph of MN (β, h) (at a xed temperature T = β −1 ) with the function
aN −1 + M (β, 0) we have the data reported in (Figure 3). The comparison shows a good
agreement between numerical and analytical estimates in the low temparature regime.
A symilar check, with similar results, has been performed also for susceptibility, internal
energy and specic heat.
Figure 3: Comparison between the coecient a = a(T ) calculated theoretically and by simulations
(2) We start by dening Ising models on nite graphs. Consider a random graph sequence
{GN }N ≥1 , where GN = (VN , EN ), with vertex set VN = [N ] = {1, ..., N } and some
random edge set EN . To each vertex i ∈ [N ] we assign an Ising spin σi = ±1. A
conguration of spins is denoted by σ = {σi : i ∈ [N ]}. The Ising model on GN is then
dened by the Boltzmann distribution
µgN (σ) =
h P
i
P
exp β (i,j)∈eN σi σj + B i∈[N ] σi
ZgN (β, B)
,
(4)
where, β ≥ 0 is the inverse temperature and B is the external magnetic eld. The random
nature of the model is reected by the fact that the probability measure (4), depending
on a random graph is, itself, a random object i.e. a random measure. The partition
function ZN (β, B) i.e. the normalization factor of Boltzmann weights, is:

ZgN (β, B) =
X
exp β
σ∈ΩN

X
(i,j)∈eN
σi σj + B
X
σi  .
(5)
i∈[N ]
In order to state our theorems, we need to dene the random set up, as follows:
1. GN = (VN , EN ) is a random graph with law QN on space GN of graphs with N
vertices, where VN = [N ] is the vertex set and EN = (EN (i, j))1≤i<j≤N is the edge
set.
2. For a given realization gN = (VN , eN ) of the random graph:
σ = {σ1 , ..., σN } is a spin conguration with law µgN on space of spin congurations
ΩN with N sites.
3. (GN ⊗ ΩN ) is the product space with law PN = QN ⊗ µgN
4
We dene
The pressure per particle: ψN (β, B) = N1 log ZgN (β, B)
P
The magnetization per vertex: MN (β, B) = N1 i∈[N ] µgN (σi )
The susceptibility : χN (β, B) =
1
N
P
(i,j)∈EN
[µgN (σi σj ) − µgN (σi )µgN (σj )] =
∂
∂B MN (β, B)
By [1] we know that
For all β > 0 and for all B ∈ R, the thermodynamic limit of the pressure exists
Φ(β, B) := lim ψN (β, B)
N →∞
For all β > 0, B 6= 0 and for all 0 < β < βc , B = 0, the thermodynamic limit of
magnetization exists and is given by
M (β, B) := lim MN (β, B) =
N →∞
∂
Φ(β, B)
∂B
For all β > 0, B 6= 0 and for all 0 < β < βc , B = 0, the thermodynamic limit of
susceptibility exists and is given by
χ(β, B) := lim χN (β, B) =
N →∞
∂2
Φ(β, B)
∂B 2
On the basis of the previous results we have proven the LLN Theorem with respect to
the random measure µgN , i.e.:
Theorem (LLN)
For all β > 0, B 6= 0 and for all 0 < β < βc , B = 0, for all sequence (gN )N ∈N :
PN
i=1
N
4
σi
P
−→ M =
∂Φ
∂B
w.r.t. µgN ,
N →∞
Future perspectives
(a) Proof of the LLN Theorem with respect to the measure PN .
(b) Proof of the Central Limit Theorem with respect to the measure µgN and with respect
to the measure PN . For all β > 0, B 6= 0 and for all 0 < β < βc , B = 0:
P
N
σ
−
µ
σ
i
g
i
N
i=1
i=1
D
√
−→ N (0, 1)
N σµgN
w.r.t. µgN ,
N →∞
P
N
σ
−
P
σ
i
N
i
i=1
i=1
D
√
−→ N (0, 1)
N σPN
w.r.t. PN ,
N →∞
PN
PN
(c) Study of the breakdown of the Central Limit Theorem at the critical point β = βc .
(d) Study of the nite volume eects in the approach to the innite volume solution of the
Ising model on random graphs [1] and [2].
(e) Study of the law of the random cavity elds at low temperatures.
5
References
[1]
, Ising models
, Journal of Statistical Physics (2010), 141, 638-660
S. Dommers, C. Giardinà and R. van der Hofstad
on power-law
random graphs
[2]
[3]
[4]
[5]
,
A. Dembo and A. Montanari
Probab. (2010) 20, 565-592.
Ising models on locally tree-like graph
, Ann. Appl.
R.S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Springer-Verlag, New
York, (1985).
C.J. Thompson
York, (1972).
,
Mathematical Statistical Mechanics
K. Binder and D.W. Heermann, Monte
Springer-Verlag, Berlin Heidelberg, (1988).
6
, The Macmillan Company, New
,
Carlo Simulation in Statistical Physics