Schramm-Loewner Evolution Introduction

In this introductory chapter, we look at iterations of conformal maps, random processes, such as random walks, and statistical physics and establish some connections.Peer reviewe

Random curves, scaling limits and Loewner evolutions

61 pages, 26 figuresIn this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally- invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves; moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.Peer reviewe

Configurations of FK Ising interfaces and hypergeometric SLE

In this paper, we show that the interfaces in the FK Ising model at criticality in a domain with 4 marked boundary points and wired-free-wired-free boundary conditions conditioned on a specific internal arc configuration of interfaces converge in the scaling limit to the hypergeometric SLE (hSLE). The arc configuration consists of a pair of interfaces and the scaling limit of their joint law can be described by an algorithm to sample the pair from an hSLE curve and a chordal SLE (in a random domain defined by the hSLE).Peer reviewe

Convergence of Ising interfaces to Schramm's SLE curves

We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter $\kappa=3$ and $\kappa=16/3$ respectively.Comment: 7 page

Satunnaiset tason käyrät ja niiden skaalausrajat

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.Tämä väitöskirja käsittelee satunnaisia käyriä tasossa ja erityisesti käyriä, jotka ovat rajapintoja tilastollisen fysiikan hilamalleissa. Käyrien ja pintojen satunnaisgeometria on viime aikoina ollut erittäin suuren mielenkiinnon kohteena matemaattisessa tilastollisessa fysiikassa. Erittäin tärkeä osa tilastollista fysiikkaa ovat hilamallit. Tarkasteltaessa tällaista mallia voidaan määrittää rajapintoja, jotka erottavat kaksi faasia toisistaan. Esimerkiksi magneettista ainetta mallintaessa rajapinta erottaa kaksi aluetta toisistaan siten, että kummassakin alueessa alkeismagneetit ovat asettautuneet yhdensuuntaisiksi, mutta toisessa alueessa päinvastaiseen suuntaan kuin toisessa. Tason hilamalleissa rajapinnat ovat käyriä, eli siis oikeammin rajakäyriä, ja kolmiulotteisen avaruuden malleissa pintoja. Tavoitteena on tutkia käyrän tai käyräperheen luonnollista jatkumorajaa, jota kutsutaan skaalausrajaksi. Kun mallin parametrit on valittu juuri tietyllä tavalla, systeemi on ns. kriittinen. Tyypillistä kriittiselle systeemille on skaalainvarianssi. Rajakäyrä katsottuna eri skaaloista näyttää tilastollisesti samalta, eli se on satunnainen fraktaali. Varsin yleisesti kriittissä systeemeissä käyrän skaalausraja voi olla vain joku SLE-käyristä (engl. Schramm-Loewner evolution). Tämä Oded Schrammin määrittelemä perhe satunnaiskäyriä on johtanut merkittäviin edistymiseen ymmärryksessä kaksiulotteisista kriittisistä systeemeistä. Tämän väitöskirjan kaksi ensimmäistä artikkelia tarkastelevat SLE-käyrien ominaisuuksia ja erityisesti niiden symmetrioita, ja niissä molemmissa kehitetään uusi tapa laskea luonnolliseen SLE-dataan liittyviä suureita. Ensimmäinen menetelmä on algebrallisempi käyttäen Virasoro-algebran konkreettista esitystä ja mahdollistaa paremmin symmetrioiden havaitsemisen. Toinen menetelmä on mahdollistaa hieman paremmin suureiden laskemisen ja tarjoaa uuden tavan tarkastella koko SLE-käyrän jakaumaa stationaarisena jakaumana. Kolmannessa artikkelissa käsitellään alan yhtä keskeisimmistä tavoitteista. Lähtien yksinkertaisesta estimaatista todistetaan tarvittavat ominaisuudet hilamallin käyrälle siten, että skaalausraja on olemassa, ja että skaalausrajaa voidaan kuvata Loewnerin yhtälöllä

Peltomaiden kalkitustarve ja kalkituksen vaikutus viljan ja nurmen satoon

vokKirjasto Aj-kEffect of liming on yield of cereals and gras

Effect of Botulinum Toxin Injection on EMG Activity and Bite Force in Masticatory Muscle Disorder: A Randomized Clinical Trial

Botulinum toxin type A (BoNT-A) is increasingly used in treating masticatory muscle pain disorder; however, safe doses and reinjection intervals still need to be established. The purpose of this randomized clinical trial was to evaluate the degree and duration of the impairment of masticatory muscle performance. Fifty-seven subjects were randomly divided into two groups: one of which received BoNT-A first (n = 28) while the other received saline first (n = 29), with the cross-over being in week 16, and a total follow-up period of 32 weeks. A total dose of 50 U of BoNT-A was injected in the masseter and temporal muscles bilaterally. Electromyographic (EMG) activity and bite forces were assessed. A significant reduction in EMG activity was observed up to week 18 (p ≤ 001), with total recovery at week 33. A significant reduction in maximum bite force was observed up to week 11 (p ≤ 005), with total recovery at week 25. In conclusion, when treating masticatory muscle pain disorder with 50 U of BoNT-A, a reinjection interval of 33 weeks can be considered safe since the recovery of muscle function occurs by that time