arXiv subjects

Michael Aizenman

Publications and source records attributed to Michael Aizenman.

58 records · Page 4Linked to original sources

arXiv1998

Continuum Limits for Critical Percolation and Other Stochastic Geometric Models

The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the scaling limit we keep track of features seen on the macroscopic scale, in situations where the short--distance scale at which the system's basic variables are defined is taken to zero. Among the challenging questions are the construction of the limit, and the explanation of some of the emergent properties, in particular the behavior under conformal maps as discussed in [LPS 94]. A descriptive account of the project, and some related open problems, is found in ref. [A] and in [AB] (joint work with A. Burchard) where tools are developed for establishing a curve--regularity condition which plays a key role in the construction of the limit. The formulation of the scaling limit as a random Web measure permits to formulate the question of uniqueness of measure(s) describing systems of random curves satisfying the conditions of independence, Euclidean invariance, and regularity. The uniqueness question remains open; progress on it could shed light on the purported universality of critical behavior and the apparent conformal invariance of the critical measures. The random Web yields also another perspective on some of the equations of conformal field theory which have appeared in this context, such as the equation proposed by J. Cardy [C].

math-ph↗

arXiv1997

Scaling Limit for the Incipient Spanning Clusters

Scaling limits of critical percolation models show major differences between low and high dimensional models. The article discusses the formulation of the continuum limit for the former case. A mathematical framework is proposed for the direct description of the limiting continuum theory. The resulting structure is expected to exhibit strict conformal invariance, and facilitate the mathematical discussion of questions related to universality of critical behavior, conformal invariance, and some relations with a number of field theories.

cond-mat.stat-mech↗

arXiv1996

On the Number of Incipient Spanning Clusters

In critical percolation models, in a large cube there will typically be more than one cluster of comparable diameter. In 2D, the probability of $k>>1$ spanning clusters is of the order $e^{-αk^{2}}$. In dimensions d>6, when $η= 0$ the spanning clusters proliferate: for $L\to \infty$ the spanning probability tends to one, and there typically are $ \approx L^{d-6}$ spanning clusters of size comparable to $|\C_{max}| \approx L^4$. The rigorous results confirm a generally accepted picture for d>6, but also correct some misconceptions concerning the uniqueness of the dominant cluster. We distinguish between two related concepts: the Incipient Infinite Cluster, which is unique partly due to its construction, and the Incipient Spanning Clusters, which are not. The scaling limits of the ISC show interesting differences between low (d=2) and high dimensions. In the latter case (d>6 ?) we find indication that the double limit: infinite volume and zero lattice spacing, when properly defined would exhibit both percolation at the critical state and infinitely many infinite clusters.

cond-mat↗

arXiv1993

Geometric Aspects of Quantum Spin States

A number of interesting features of the ground states of quantum spin chains are analized with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU($2S+1$)-invariant quantum spin-$S$ chains with the interaction $-P^{(0)}$, where $P^{(0)}$ is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Néel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the $-P^{(0)}$ spin-$S$ systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.

cond-mat↗