2,415 research outputs found
Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction
For random collections of self-avoiding loops in two-dimensional domains, we
define a simple and natural conformal restriction property that is
conjecturally satisfied by the scaling limits of interfaces in models from
statistical physics. This property is basically the combination of conformal
invariance and the locality of the interaction in the model. Unlike the Markov
property that Schramm used to characterize SLE curves (which involves
conditioning on partially generated interfaces up to arbitrary stopping times),
this property only involves conditioning on entire loops and thus appears at
first glance to be weaker.
Our first main result is that there exists exactly a one-dimensional family
of random loop collections with this property---one for each k in (8/3,4]---and
that the loops are forms of SLE(k). The proof proceeds in two steps. First,
uniqueness is established by showing that every such loop ensemble can be
generated by an "exploration" process based on SLE.
Second, existence is obtained using the two-dimensional Brownian loop-soup,
which is a Poissonian random collection of loops in a planar domain. When the
intensity parameter c of the loop-soup is less than 1, we show that the outer
boundaries of the loop clusters are disjoint simple loops (when c>1 there is
almost surely only one cluster) that satisfy the conformal restriction axioms.
We prove various results about loop-soups, cluster sizes, and the c=1 phase
transition.
Taken together, our results imply that the following families are equivalent:
1. The random loop ensembles traced by certain branching SLE(k) curves for k
in (8/3, 4].
2. The outer-cluster-boundary ensembles of Brownian loop-soups for c in (0,
1].
3. The (only) random loop ensembles satisfying the conformal restriction
axioms.Comment: This 91 page-long paper contains the previous versions (v2) of both
papers arxiv:1006.2373 and arxiv:1006.2374 that correspond to Part I and Part
II of the present paper. This merged longer paper is to appear in Annals of
Mathematic
Universality in the 2D Ising model and conformal invariance of fermionic observables
It is widely believed that the celebrated 2D Ising model at criticality has a
universal and conformally invariant scaling limit, which is used in deriving
many of its properties. However, no mathematical proof of universality and
conformal invariance has ever been given, and even physics arguments support (a
priori weaker) M\"obius invariance. We introduce discrete holomorphic fermions
for the 2D Ising model at criticality on a large family of planar graphs. We
show that on bounded domains with appropriate boundary conditions, those have
universal and conformally invariant scaling limits, thus proving the
universality and conformal invariance conjectures.Comment: 52 pages, 11 figures. Minor changes + two important ones: a) Section
3.4 (a priori Harnack principle for H) added; b) Section 5 (spin-observable
convergence) simplified and rewritten (boundary Harnack principle added,
solution in the half-plane simplified
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
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