5 research outputs found
Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
A statistical mechanics argument relating partition functions to martingales
is used to get a condition under which random geometric processes can describe
interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs
to satisfy this condition leads to some natural processes, which we study in
this note. We give examples of such multiple SLEs and discuss how a choice of
conformal block is related to geometric configuration of the interfaces and
what is the physical meaning of mixed conformal blocks. We illustrate the
general ideas on concrete computations, with applications to percolation and
the Ising model.Comment: 40 pages, 6 figures. V2: well, it looks better with the addresse
Virasoro Module Structure of Local Martingales of SLE Variants
Martingales often play an important role in computations with Schramm-Loewner
evolutions (SLEs). The purpose of this article is to provide a straightforward
approach to the Virasoro module structure of the space of local martingales for
variants of SLEs. In the case of ordinary chordal SLE, it has been shown in
Bauer & Bernard: Phys.Lett.B 557 that polynomial local martingales form a
Virasoro module. We will show for more general variants that the module of
local martingales has a natural submodule M that has the same interpretation as
the module of polynomial local martingales of chordal SLE, but it is in many
cases easy to find more local martingales than that. We discuss the
surprisingly rich structure of the Virasoro module M and construction of the
``SLE state'' or ``martingale generating function'' by Coulomb gas formalism.
In addition, Coulomb gas or Feigin-Fuchs integrals will be shown to
transparently produce candidates for multiple SLE pure geometries.Comment: 48 pages, 3 figures. v4: Completely reorganized, with new results,
erroneous corollary 4 (in v3) correcte
SLE local martingales in logarithmic representations
A space of local martingales of SLE type growth processes forms a
representation of Virasoro algebra, but apart from a few simplest cases not
much is known about this representation. The purpose of this article is to
exhibit examples of representations where L_0 is not diagonalizable - a
phenomenon characteristic of logarithmic conformal field theory. Furthermore,
we observe that the local martingales bear a close relation with the fusion
product of the boundary changing fields.
Our examples reproduce first of all many familiar logarithmic representations
at certain rational values of the central charge. In particular we discuss the
case of SLE(kappa=6) describing the exploration path in critical percolation,
and its relation with the question of operator content of the appropriate
conformal field theory of zero central charge. In this case one encounters
logarithms in a probabilistically transparent way, through conditioning on a
crossing event. But we also observe that some quite natural SLE variants
exhibit logarithmic behavior at all values of kappa, thus at all central
charges and not only at specific rational values.Comment: 40 pages, 7 figures. v3: completely rewritten, new title, new result
LERW as an example of off-critical SLEs
Two dimensional loop erased random walk (LERW) is a random curve, whose
continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter
kappa=2. In this article we study ``off-critical loop erased random walks'',
loop erasures of random walks penalized by their number of steps. On one hand
we are able to identify counterparts for some LERW observables in terms of
symplectic fermions (c=-2), thus making further steps towards a field theoretic
description of LERWs. On the other hand, we show that it is possible to
understand the Loewner driving function of the continuum limit of off-critical
LERWs, thus providing an example of application of SLE-like techniques to
models near their critical point. Such a description is bound to be quite
complicated because outside the critical point one has a finite correlation
length and therefore no conformal invariance. However, the example here shows
the question need not be intractable. We will present the results with emphasis
on general features that can be expected to be true in other off-critical
models.Comment: 45 pages, 2 figure