Docente

RICCI TERSENGHI FEDERICO
(programma)
Intro on disordered systems I plan to study in these lectures: disordered ferromagnets, spin glasses and structural glasses (selfinduced disorder). Applications in computer science: constraint satisfaction problems (with a brief reminder on complexity theory) and Bayesian inference problems.
Reminder on probability theory. Random variables, expectation, variance. Shannon entropy. KullbackLeibler divergence. Correlated variables: conditional probability, conditional entropy and mutual information. Law of large numbers, central limit theorem. Theory of large deviations.
The Ising model: discussion on the connection between the physical behavior and the topology (short range, long range, infinite range). Upper and lower critical dimensions. The solution to the Ising model in D=1 and no external field via the high temperature expansion. Correlation functions and correlation length.
Solution of the D=1 Ising model via the transfer matrix formalism. Magnetic susceptibility and its connection to correlation function via the fluctuationdissipation relation. Decay rate in the correlation function and in the joint probability distribution. Random signs in the couplings and gauge transformation. Non uniform fields.
CurieWeiss model: computation of the thermodynamical freeenergy, via the freeenergy at fixed magnetization, f(m). Physical meaning of f(m). Phase diagram in the plane (T,h): spinodal lines and regions of phase coexistence. Metastable states.
Lower critical dimension for Ising and O(n1) models. How to compute the stability of the ground state via simple compact excitations.
First order and second order phase transitions in the CurieWeiss model. Critical behavior around the second order critical point. Critical exponents and universality. A model with a first order transition in temperature: the fully connected 3spin ferromagnetic Ising model.
Stability of first and second order phase transitions under the application of an external field.
A discussion of 3 concepts that often come together: thermodynamic phase transition, spontaneous symmetry breaking, criticality.
Ising model in D=2. Proof that a ferromagnetic phase exists a low enough temperature. Disordered models in D=2: diluted models, RFIM and spin glasses. Estimating the amount of frustration computing frustrated plaquettes. Mattis model has random couplings but no frustration. Computing the ground state on a D=2 spin glass is a matching problem among frustrated plaquettes. Annealed and quenched averages over the disorder.
Selfaverageness (first and second moment of the partition function, selfaveraging in finite dimensional short range models). Diluted Ising models: qualitative phase diagram from critical point of the pure model and from the percolation point. Bound on the critical line in the (p,T) plane from the concavity of in the couplings. Discussion of the stability of the pure model fixed points under RG and the existence of random RG fixed points. The Harris criterion and its extension to correlated disorder.
The paramagnetic phase of models with disorder is not simple. Singularities in the pure model become points of nonanalyticity in the model with disorder. Griffiths singularities. Dynamics in pure models: phase ordering kinetics. The quenching protocol, the observation of a growing correlation length. Quantifying the correlation length via the correlation function. Scaling hypothesis for the correlation function.
Timedependent GinnzburgLandau (TDGL) equation for nonconserved phase ordering dynamics. Scaling hypothesis. Stationary solution for a single domain wall. Solution for a spherical droplet: relation between domain wall velocity and domain wall curvature. A first estimate of the growing law for the timedependent spatial correlation length from the law of annihilation of spherical domains. Solution to the TDGL equation by linearization of the nonlinear term (an approximation which is exact for the O(n) model).
Discussion on the aging solution in the outofequilibrium dynamics of the O(n) model. Relaxation in the 1D ising model: solution via the master equation and via the defect dynamics. Discussion of similarities between relaxation towards equilibrium and decorrelation at equilibrium. Critical dynamics: divergence of timescales and lengthscales at a critical point. Critical dynamical exponent and its scaling form.
Relaxation dynamics in the diluted ferromagnet for TTc(p=1). Estimation of clusters maximizing the correlation at any given time. Anomalous large timescale for the correlation decay and stretched exponential decay. Relaxation dynamics in the diluted ferromagnet for Tc(p)0 and thermodynamical limit. Replica symmetric (RS) ansatz and stability criterion. The replicated freeenergy for the coupled system and the ansatz for the one step of replica symmetry breaking (1RSB). From the replicated freeenergy to the phase diagram in the (m,T) plane and the corresponding phase transitions in the pspin model. Connections with the dynamical behavior.
Overlaps among states and their probability distribution P(q) as the order parameter of the spin glass phase transition. Physical meaning of the matrix Q extremizing the replicated freeenergy in terms of the P(q). Phase diagram of the SK model and schematic behavior of P(q).
Detailed discussion of the P(q) as the order parameter of the SK model: q_EA(T) and q_min(h). Susceptibilities in presence of many states: global response versus local response within a single state, and their dynamical counterparts. Linear susceptibilities in spin glasses. Comparison of fieldcooled and zerofieldcooled susceptibilities measured in experiments with those computed in the SK model. Spin glass susceptibility.
A short introduction to random graphs: definition of the ensembles G(N,M) and G(N,p), distribution of degrees, size of loops, local convergence to a tree. Factor graphs to represent multivariables interactions. Models defined on factor graphs (graphical models). Computing marginals is as hard as computing the partition function. The Bethe approximation as a factorization involving higher order statistics. The Bethe approximation is exact on trees.
The Bethe freeenergy and saddle point equations to extremize it. Belief Propagation an iterative algorithm for solving the Bethe saddle point equations. Fixed point of BP corresponds to stationary points of the Bethe free energy (under the normalization and consistency constraints).
Computing distant correlations via linear response within the Bethe approximation. Connecting the stability of the BP fixed point with the susceptibility. An explicit example, the ferromagnetic model on a random regular graph: the saddle point equations, the analytic expression for the critical point and the freeenergy. Limits of the Bethe approximation (aka RS cavity method). On the use of the Bethe approximation for models defined on graphs with short loops (e.g. regular lattices). Comparison of nMF, Bethe and higher order approximations (Cluster Variational Methods) applied to the 2D Ising model.
Computing correlations in the Ising model on a random regular graph. Comparing the diverging susceptibility on finite dimensional spaces and on a random graph. Qualitative behaviour of the stability parameter lambda in ferromagnetic models and in spin glass models. TAP equations as the limit of the BP equations for models defined on dense graphs. Using the Bethe approximation for estimating averages over the ensemble or on a typical very large sample. Population Dynamics for computing the distributions of cavity marginals.
Different longrange correlations leads to different diverging susceptibilities and different physical behaviors. Pointtoset correlation function. Schematic phase diagram RSd1RSBs1RSB. Random XORSAT, a simple model with such a phase diagram: marginals, cavity marginals and BP equations. Dynamical phase transition and the existence of nontrivial BP fixed points. Solutions on the 2core corresponds to clusters of solutions. Algorithmic consequences.
Note del corso
Marc Mézard and Andrea Montanari, Information, Physics, and Computation (Oxford University Press, 2009)
Colin J. Thompson, Mathematical Statistical Mechanics (Princeton University Press, 1979)
A. Weinrib e B. Halperin, Critical phenomena in systems with longrangecorrelated quenched disorder, Phys. Rev. B 27 (1983) 413
R. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23 (1969) 17
J. Froelich, Mathematical Aspects of the Physics of Disordered Systems, in Critical Phenomena, Random Systems, Gauge Theories, edited by K. Osterwalder and R. Stora, Les Houches, session XLIII, 1984 (Elsevier, 1986)
Alan J. Bray, Theory of phaseordering kinetics, Advances in Physics 43 (1994) 357
Uwe C. Täuber, Critical Dynamics, A field theory approach to equilibrium and nonequilibrium scaling behavior (Cambridge University Press, 2014)
T. Nattermann, in Spin Glasses and Random Fields edited by A.P. Young, p. 277 (World Scientific, Singapore 1998)
J. Cardy, Scaling and Renormalization in Statistical Physics in Cambridge Lecture Notes in Physics (Cambridge University Press, 1996)
F. Zamponi, Mean field theory of spin glasses, arXiv:1008.4844
T. Castellani and A. Cavagna, SpinGlass Theory for Pedestrian, arXiv:condmat/0505032
F. RicciTersenghi, The Bethe approximation for solving the inverse Ising problem: a comparison with other inference methods, J. Stat. Mech. P08015 (2012), arXiv:1112.4814 (sections 1 and 2)
