Spin Glass Computations and Ruelle's Probability Cascades

We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure.Comment: 20 page

Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming issue of Probab. Theory Relat. Field

Proof of Rounding by Quenched Disorder of First Order Transitions in Low-Dimensional Quantum Systems

We prove that for quantum lattice systems in d<=2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T=0. For systems with continuous symmetry the statement extends up to d<=4 dimensions. This establishes for quantum systems the existence of the Imry-Ma phenomenon which for classical systems was proven by Aizenman and Wehr. The extension of the proof to quantum systems is achieved by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.Comment: This article presents the detailed derivation of results which were announced in Phys. Rev. Lett. 103 (2009) 197201 (arXiv:0907.2419). v3 incorporates many corrections and improvements resulting from referee comment

The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr package

The Effect of Crystallization on the Pulsations of White Dwarf Stars

We consider the pulsational properties of white dwarf star models with temperatures appropriate for the ZZ Ceti instability strip and with masses large enough that they should be substantially crystallized. Our work is motivated by the existence of a potentially crystallized DAV, BPM 37093, and the expectation that digital surveys in progress will yield many more such massive pulsators. A crystallized core makes possible a new class of oscillations, the torsional modes, although we expect these modes to couple at most weakly to any motions in the fluid and therefore to remain unobservable. The p-modes should be affected at the level of a few percent in period, but are unlikely to be present with observable amplitudes in crystallizing white dwarfs any more than they are in the other ZZ Ceti's. Most relevant to the observed light variations in white dwarfs are the g-modes. We find that the kinetic energy of these modes is effectively excluded from the crystallized cores of our models. As increasing crystallization pushes these modes farther out from the center, the mean period spacing between radial overtones increases substantially with the crystallized mass fraction. In addition, the degree and structure of mode trapping is affected. The fact that some periods are strongly affected by changes in the crystallized mass fraction while others are not suggests that we may be able to disentangle the effects of crystallization from those due to different surface layer masses.Comment: 18 pages, 5 figures, accepted on 1999 July 2 for publication in the Astrophysical Journa

Short-range spin glasses and Random Overlap Structures

Properties of Random Overlap Structures (ROSt)'s constructed from the Edwards-Anderson (EA) Spin Glass model on $\Z^d$ with periodic boundary conditions are studied. ROSt's are $\N\times\N$ random matrices whose entries are the overlaps of spin configurations sampled from the Gibbs measure. Since the ROSt construction is the same for mean-field models (like the Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the setup is a good common ground to study the effect of dimensionality on the properties of the Gibbs measure. In this spirit, it is shown, using translation invariance, that the ROSt of the EA model possesses a local stability that is stronger than stochastic stability, a property known to hold at almost all temperatures in many spin glass models with Gaussian couplings. This fact is used to prove stochastic stability for the EA spin glass at all temperatures and for a wide range of coupling distributions. On the way, a theorem of Newman and Stein about the pure state decomposition of the EA model is recovered and extended.Comment: 27 page

Chaotic temperature dependence at zero temperature

We present a class of examples of nearest-neighbour, boubded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension

Roles of NR2A and NR2B in the development of dendritic arbor morphology in vivo

NMDA receptors (NMDARs) are important for neuronal development and circuit formation. The NMDAR subunits NR2A and NR2B are biophysically distinct and differentially expressed during development but their individual contribution to structural plasticity is unknown. Here we test whether NR2A and NR2B subunits have specific functions in the morphological development of tectal neurons in living Xenopus tadpoles. We use exogenous subunit expression and endogenous subunit knockdown to shift synaptic NMDAR composition toward NR2A or NR2B, as shown electrophysiologically. We analyzed the dendritic arbor structure and found evidence for both overlapping and distinct functions of NR2A and NR2B in dendritic development. Control neurons develop regions of high local branch density in their dendritic arbor, which may be important for processing topographically organized inputs. Exogenous expression of either NR2A or NR2B decreases local branch clusters, indicating a requirement for both subunits in dendritic arbor development. Knockdown of endogenous NR2A reduces local branch clusters, whereas knockdown of NR2B has no effect on branch clustering. Analysis of the underlying branch dynamics shows that exogenous NR2B-expressing neurons are more dynamic than control or exogenous NR2A-expressing neurons, demonstrating subunit-specific regulation of branch dynamics. Visual experience-dependent increases in dendritic arbor growth rate seen in control neurons are blocked in both exogenous NR2A- and NR2B-expressing neurons. These experiments indicate that NR2A and NR2B have subunit-specific properties in dendritic arbor development, but also overlapping functions, indicating a requirement for both subunits in neuronal development

Ground State Structure in a Highly Disordered Spin Glass Model

We propose a new Ising spin glass model on $Z^d$ of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite volume) ground states for this model can be related to invasion percolation with the number of ground states identified as $2^{\cal N}$, where ${\cal N} = {\cal N}(d)$ is the number of distinct global components in the ``invasion forest''. We prove that ${\cal N}(d) = \infty$ if the invasion connectivity function is square summable. We argue that the critical dimension separating ${\cal N} = 1$ and ${\cal N} = \infty$ is $d_c = 8$. When ${\cal N}(d) = \infty$, we consider free or periodic boundary conditions on cubes of side length $L$ and show that frustration leads to chaotic $L$ dependence with {\it all} pairs of ground states occuring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.Comment: LaTex fil

Numerical Simulations of the 4D Edwards-Anderson Spin Glass with Binary Couplings

We present numerical results that allow a precise determination of the transition point and of the critical exponents of the 4D Edwards-Anderson Spin Glass with binary quenched random couplings. We show that the low T phase undergoes Replica Symmetry Breaking. We obtain results on large lattices, up to a volume $V=10^4$: we use finite size scaling to show the relevance of our results in the infinite volume limit.Comment: 18 pages + 17 figures, revised bibliography and minor typos. Added Journal Re