On the Parisi-Toulouse hypothesis for the spin glass phase in mean-field theory

We consider the spin-glass phase of the Sherrington-Kirkpatrick model in the presence of a magnetic field. The series expansion of the Parisi function $q(x)$ is computed at high orders in powers of $\tau=T_c-T$ and $H$. We find that none of the Parisi-Toulouse scaling hypotheses on the $q(x)$ behavior strictly holds, although some of them are violated only at high orders. The series is resummed yielding results in the whole spin-glass phase which are compared with those from a numerical evaluation of the $q(x)$. At the high order considered, the transition turns out to be third order on the Almeida-Thouless line, a result which is confirmed rigorously computing the expansion of the solution near the line at finite $\tau$. The transition becomes smoother for infinitesimally small field while it is third order at strictly zero field.Comment: 6 pages, 2 figure

Beyond the Sherrington-Kirkpatrick Model

The state of art in spin glass field theory is reviewed.Comment: contribution to the volume "Spin Glasses and Random Fields", ed. P. Young, World Scientific. Latex file and lprocl.sty (style-file). 41 pages, no figure

On Ward-Takahashi identities for the Parisi spin glass

The introduction of ``small permutations'' allows us to derive Ward-Takahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes. The next identities relate components of (the Replica Fourier Transformed) 3-point function to overlap derivatives of the 2-point function (inverse propagator). A jump in this last function is exhibited, when its two overlaps are crossing each other, in the special simpler case where one of the cross-overlaps is maximal.Comment: this new version includes acknowledgements to funding agencie

Stability of the Mezard-Parisi solution for random manifolds

The eigenvalues of the Hessian associated with random manifolds are constructed for the general case of $R$ steps of replica symmetry breaking. For the Parisi limit $R\to\infty$ (continuum replica symmetry breaking) which is relevant for the manifold dimension $D<2$, they are shown to be non negative.Comment: LaTeX, 15 page

Random field Ising model: dimensional reduction or spin-glass phase?

The stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by M\'ezard and Young, is naturally expressed via Legendre transforms, stability being then associated with the non-negativeness of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is found that the signal for the occurrence of the SG transition will manifest itself in free-energy {\sl fluctuations\/} only, and not in the free energy itself. Eigenvalues of the inverse SG susceptibility matrix is then approached by the Rayleigh Ritz method which provides an upper bound. Coming from the paramagnetic phase {\sl on the Curie line,\/} one is able to use a virial-like relationship generated by scaling the {\sl single\/} unit length $(D<6;$ in higher dimension a new length sets in, the inverse momentum cut off). Instability towards a SG phase being probed on pairs of {\sl distinct\/} replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is {\sl attractive\/} (at least for small values of the parameter $g\bar \Delta ,$ $g$ the coupling and $\bar \Delta$ the effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\ associated with replica pairs (negative eigenvalues) provide the instability signature. {\sl Away from the Curie line\/}, the attraction is damped out till the SG transition line is reached and paramagnetism restored. In $D<6,$ the SG transition always precedes the ferromagnetic one, thus the domain in dimension where standard dimensional reduction would apply (on the Curie line) shrinks to zero.Comment: te

Unaccompanied Minors? An analysis of the legal situation of abandoned children born in Hungary

The MPC is co-financed by the European University Institute and the European Union.In recent years changes in Hungarian citizenship policy and legislation have aroused public interest. The efforts of the Hungarian government to facilitate the naturalisation of ethnic Hungarians particularly encountered esistance from neighbouring countries,1 and was also viewed critically by some scholars.2 At the same time, the issue of unaccompanied minors has been high on the political agenda in EU Member States, including Hungary. Various EU institutions and bodies have commissioned studies and reports to analyse the situation in the European Union3 and an Action Plan was launched in order to ensure greater coherence and cooperation and to improve the protection offered to this vulnerable group.4 Nevertheless, a group of unaccompanied minor children, who do not fit into the traditional definition of unaccompanied minors in Europe, has been neglected. These children were born in Hungary of a foreign national, but of a Hungarian speaking and presumably ethnic Hungarian mother who subsequently abandoned the child in hospital shortly after birth. Despite liberal citizenship policy and an existing legal framework for the protection of unaccompanied minors, these children do not, for various reasons, obtain any nationality at or after birth and remain in a legal limbo for many months or even years. The aim of this paper is to explore the legal situation of these children in three areas: citizenship, immigration status and reception and care, and to analyse to what extent the current practices of the Guardianship Office and the Office of Immigration and Nationality is in compliance with Hungary’s international legal obligations, with Community law and, indeed, with domestic law. Particular attention will be paid to the obligations of Hungary as set out in the Convention of the Rights of the Child, the Convention relating to the Status of Stateless Persons and the Convention on the Reduction of Statelessness. As an unclear citizenship status constitutes the main reason for their peculiar situation, we will also look at the possibility of granting Hungarian citizenship or stateless status

Replica Fourier Transforms on Ultrametric Trees, and Block-Diagonalizing Multi-Replica Matrices

The analysis of objects living on ultrametric trees, in particular the block-diagonalization of 4-replica matrices $M^{\alpha \beta ; \gamma \delta}$, is shown to be dramatically simplified through the introduction of properly chosen operations on those objects. These are the Replica Fourier Transforms on ultrametric trees. Those transformations are defined and used in the present work.Comment: Latex file, 14 page

Reparametrization invariance: a gauge-like symmetry of ultrametrically organised states

The reparametrization transformation between ultrametrically organised states of replicated disordered systems is explicitly defined. The invariance of the longitudinal free energy under this transformation, i.e. reparametrization invariance, is shown to be a direct consequence of the higher level symmetry of replica equivalence. The double limit of infinite step replica symmetry breaking and n=0 is needed to derive this continuous gauge-like symmetry from the discrete permutation invariance of the n replicas. Goldstone's theorem and Ward identities can be deduced from the disappearence of the second (and higher order) variation of the longitudinal free energy. We recall also how these and other exact statements follow from permutation symmetry after introducing the concept of "infinitesimal" permutations.Comment: 16 pages, 3 figure