Spectral renormalization group theory on nonspatial networks

Physical Review E, 2005

We have defined a new type of clustering scheme preserving the connectivity of the nodes in network ignored by the conventional Migdal-Kadanoff bond moving process. Our new clustering scheme performs much better for correlation length and dynamical critical exponents in high dimensions, where the conventional Migdal-Kadanoff bond moving scheme breaks down. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising Model to be z = 2.13 and z = 2.09, respectively at pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behaviour for randomly bond diluted lattices in d=2 and 3, in the light of this new transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law degree distributions, both in the pure and random cases.

Communications in Mathematical Physics, 1983

New methods are developed for the study of the Kadanoff-Wilson renormalization group for critical lattice systems of unbounded spins. The methods are based on a combination of expansion and analyticity techniques and are applied to a nonlocal hierarchical model of the dipole gas. They remove the main obstacle against the use of the block spin strategy in more realistic models such as φ* 9 the dipole gas and the anharmonic crystal.

Physical Review B, 1997

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.

Journal of Mathematical Physics, 2003

We model quantum space–time on the Planck scale as dynamical networks of elementary relations or time dependent random graphs, the time dependence being an effect of the underlying dynamical network laws. We formulate a kind of geometric renormalization group on these (random) networks leading to a hierarchy of increasingly coarse-grained networks of overlapping lumps. We provide arguments that this process may generate a fixed limit phase, representing our continuous space–time on a mesoscopic or macroscopic scale, provided that the underlying discrete geometry is critical in a specific sense (geometric long range order). Our point of view is corroborated by a series of analytic and numerical results, which allow us to keep track of the geometric changes, taking place on the various scales of the resolution of space–time. Of particular conceptual importance are the notions of dimension of such random systems on the various scales and the notion of geometric criticality.

2020

Systems with lattice geometry can be renormalized exploiting their embedding in metric space, which naturally defines the coarse-grained nodes. By contrast, complex networks defy the usual techniques because of their small-world character and lack of explicit metric embedding. Current network renormalization approaches require strong assumptions (e.g. community structure, hyperbolicity, scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Here we introduce a graph renormalization scheme valid for any hierarchy of coarse-grainings, thereby allowing for the definition of `block nodes' across multiple scales. This approach reveals a necessary and specific dependence of network topology on an additive hidden variable attached to nodes, plus optional dyadic factors. Renormalizable networks turn out to be consistent with a unique specification of the fitness model, while they are incompatible with preferential attachment, the configuration model...

1978

This thesis is divided into two parts. In Part I, we give an explicit construction for a class of lattices with effectively non-integral dimensionality. A reasonable definition of dimensionality applicable to lattice systems is proposed. The construction is illustrated by several examples. We calculate the effective dimensionality of some of these lattices. The attainable values of the dimensionality d, using our construction, are densely distributed in the interval 1 The variation of critical exponents with dimensionality is studied for a variety of Hamiltonians. It is shown that the critical exponents for the spherical model, for all d, agree with the values derived in literature using formal arguments only. We also study the critical behavior of the classical p-vector Heisenberg model and the Fortuin-Kasteleyn cluster model for lattices with d<2. It is shown that no phase transition occurs at nonzero temperatures. The renormalization procedure is used to determine the exact va...

2012

In this thesis we investigate the Renormalization Group (RG) approach in finitedimensional glassy systems, whose critical features are still not well-established, or simply unknown. We focus on spin and structural-glass models built on hierarchical lattices, which are the simplest non-mean-field systems where the RG framework emerges in a natural way. The resulting critical properties shed light on the critical behavior of spin and structural glasses beyond mean field, and suggest future directions for understanding the criticality of more realistic glassy systems. v

2008

Recently, kinds of scaling schemes for general large scale complex networks have been developed and attracted much attention. We propose a new scaling scheme named "two-sites scaling" and investigate how the degree distribution of network changes in applying the proposed scheme to various networks. Notably, the results indicate that networks constructed by BA algorithm behave differently compared with networks commonly appearing in the real world. In addition, since an iterative scaling scheme could define a new renormalizing method, we argue about using our scheme for Wilsonian renormalization group theory for general complex networks and its application to analyzing the dynamics of complex networks.

2001

We reformulate our dynamical networks, developed elsewhere, as causal sets, more properly, time dependent graphs, carrying a dynamical causal structure which is space-time dependent, thus incorporating the spirit of general relativity. In the main part of the paper we then develop a geometric renormalisation group, acting on these networks or graphs, aiming at coarse-graining the fine structure, being prevalent at or around the Planck scale. We show that repeated application of these renormalisation steps may lead to a macroscopic fixed point or attracting fixed phase, playing the role of a continuous macroscopic phase, we call space-time. We furthermore show that this renormalisation map can be understood as a physically meaningful (endo)functor in some category ‘Graphs’. To study the nature of our coarse-graining procedure in more detail, we employ the concept of network or graph dimension as an important geometric characteristic of such large irregular arrays of degrees of freedo...

Physical Review B, 1983

It is shown that some recently proposed iterative approaches to the ground state of quantum systems can be obtained as zero-temperature limits of free-energy-preserving renormalization transformations, if the perturbative expansion is handled in the appropriate way. Besides giving new insight into the ground-state approaches and their mutual relationships, these results allow us to perform consistent approximate calculations of the free energy at all temperatures and to get global descriptions of the critical properties. Free-energy and specific-heat calculations are reported for XY and Heisenberg spin- 1/2 chains and for the triangular XY model. It is also demonstrated how this extension into finite temperatures allows us to compute in a consistent way the z exponent, and to obtain substantial improvement in the numerical values for the ground-state energy density of the transverse Ising model.