2d frustrated Ising model with four phases

In this paper we consider a 2d random Ising system on a square lattice with nearest neighbour interactions. The disorder is short range correlated and asymmetry between the vertical and the horizontal direction is admitted. More precisely, the vertical bonds are supposed to be non random while the horizontal bonds alternate: one row of all non random horizontal bonds is followed by one row where they are independent dichotomic random variables. We solve the model using an approximate approach that replace the quenched average with an annealed average under the constraint that the number of frustrated plaquettes is keep fixed and equals that of the true system. The surprising fact is that for some choices of the parameters of the model there are three second order phase transitions separating four different phases: antiferromagnetic, glassy-like, ferromagnetic and paramagnetic.Comment: 17 pages, Plain TeX, uses Harvmac.tex, 4 ps figures, submitted to Physical Review

Non-Mean-Field Behavior of Realistic Spin Glasses

We provide rigorous proofs which show that the main features of the Parisi solution of the Sherrington-Kirkpatrick spin glass are not valid for more realistic spin glass models in any dimension and at any temperature.Comment: LaTeX file, 8 page

Complex Random Energy Model: Zeros and Fluctuations

The partition function of the random energy model at inverse temperature $\beta$ is a sum of random exponentials $Z_N(\beta)=\sum_{k=1}^N \exp(\beta \sqrt{n} X_k)$, where $X_1,X_2,...$ are independent real standard normal random variables (= random energies), and $n=\log N$. We study the large $N$ limit of the partition function viewed as an analytic function of the complex variable $\beta$. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $\beta$, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.Comment: 31 pages, 1 figur

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

The low-temperature free energy of the Sherrington-Kirkpatrick spin glass model

We show that, for $\beta \geq 1$, the specific free energy of the Sherrington-Kirkpatrick spin glass model is bounded by 
[se the full version ]. 
This and the derived bound for the ground-state energy improve all the recently obtained results

A stability criterion for financial markets

Using the theory of random cluster models, we give a stability criterion for financial markets with random communications between agents

LOW TEMPERATURE PHASE DIAGRAM FOR A CLASS OF FINITE RANGE INTERACTIONS ON THE PENROSE LATTICE

Recently, new alloys have been synthesized that exhibit X-ray diffraction patterns with unusual crystallographic symmetries, excluded by classical crystallography, like five- or ten-fold axes. It is shown that such patterns can be obtained by diffraction on quasiperiodic structures like the Penrose tiling of the plane. To describe the quasiperiodic lattice, we use the projection method introduced in [1]. Namely, we decompose the space R * into two orthogonal subspaces Ek and E? and denote by ssk and ss? the corresponding projections. The quasiperiodic lattice is identified to a particular discrete subset of Ek that will be constructed in the sequel. Denote by fffl1; : : : ; ffl*g an orthonormal basis of R*, by fl the unit hypercube fl = f, 2 R * : ,

Mathematical physics of disordered systems Abstracts

SIGLEAvailable from TIB Hannover: RR 3285(3)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman