Kalle Kytölä

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

On staggered indecomposable Virasoro modules

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. Moreover, the module is further assumed to have a highest weight submodule, the “left module,” and that the quotient by this submodule yields another highest weight module, the “right module.” Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centers on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are define...

Ising interfaces and free boundary conditions

The authors generalize the result of D. Chelkak et al. [C. R., Math., Acad. Sci. Paris 352, No. 2, 157{161 (2014; Zbl 06265643)] to the case when free boundary conditions enter the picture. The proof is related to the rigorous computation of a (dual) boundary conformal eld theory correlation function, which is obtained by using both resent results concerning the boundary correlation functions of the model and existing SLE results for dual models. The result is applied for the conformal invariance of crossing probabilities. Another application is the proof that the collection of the Ising model interfaces converges to the conformal loop ensemble.

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.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 paper is to exhibit examples of representations where L0 is not diagonalizable—a phenomenon characteristic of logarithmic conformal field theory. Furthermore, we observe that the local martingales bear a close relation to 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 SLEκ=6 describing the exploration path in critical percolation and its relation to 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 κ, thus at all central charges and not only at specific rational values.

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 κ=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.

SLE local martingales, reversibility and duality

We study Schramm–Loewner evolutions (SLEs) reversibility and duality using the Virasoro structure of the space of local martingales. For both problems we formulate a setup where the questions boil down to comparing two processes at a stopping time. We state algebraic results showing that local martingales for the processes have enough in common. When one has in addition integrability, the method gives reversibility and duality for any polynomial expected value.

SLE boundary visits

We study the probabilities with which chordal Schramm–Loewner evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green’s functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain–Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko–Fateev type integrals. We show how to express these integral formulas in terms of regularized real integrals, and we discuss their numerical evaluation. The results are universal in the sense that apart from an overall multiplicative constant the same formula gives the amplitude for many different formulations of the SLE boundary visit problem. The formula also applies to renormalized boundary visit probabilities for interfaces in critical lattice models of statistical mechanics: we compare the results with numerical simulations of percolation, loop-erased random walk, and Fortuin–Kasteleyn random cluster models at Q = 2 and Q = 3, and find good agreement.

Elastic manifolds in disordered environments: Energy statistics

The energy of an elastic manifold in a random landscape at T = 0 is shown numerically to obey a probability distribution that depends on the size of the box it is put into. If the extent of the spatial fluctuations of the manifold is much less than that of the system, a crossover takes place to the Gumbel distribution of extreme statistics. If they are comparable, the distributions have non-Gaussian, stretched exponential tails. The low-energy and high-energy stretching exponents are roughly independent of the internal dimension and the fluctuation degrees of freedom.