arXiv subjects

Eveliina Peltola

Publications and source records attributed to Eveliina Peltola.

At least 19 recordsLinked to original sources

arXiv2026

Complex deformations of the circle: Group cohomology and Virasoro uniformization

We approach the question of complexification of the diffeomorphism group of the circle by considering real-analytic maps from the circle into the punctured complex plane with winding number +1. Such complex deformations form an infinite-dimensional manifold with partially defined inversion and composition operations, smooth in the sense of Frölicher structures, and with Lie algebra relations at the identity given by the Witt algebra. We compute their second group cohomology with real coefficients. It is generated by two cocycles: One cocycle extending the Bott-Thurston cocycle related to the Gelf'and-Fuks cocycle of the Virasoro algebra, and another Euler class cocycle combining mean rotation and conformal radius. Complex deformations act naturally on the (infinite-dimensional) Segal moduli spaces of Riemann surfaces with enumerated and analytically parametrized boundary components, equipping them with smooth Fr{ö}licher structures. We prove a Virasoro uniformization theorem: The tangent spaces of the Segal moduli spaces are spanned by vector fields induced by the Witt algebra. Finally, we relate the actions of complex deformations to Fenchel-Nielsen coordinates and Schiffer variation on finite-dimensional moduli spaces of bordered hyperbolic surfaces with one marked point on each boundary component.

math-ph↗

arXiv2026

Universality of the conformal anomaly

The sewing operation is the fundamental algebraic structure of the (infinite-dimensional) Segal moduli spaces as the conformal cobordism category. We prove a universal property of local $\mathbb{R}_+$-extensions of the sewing operation, characterizing them by one real parameter: their central charge. Our result is an analogue of ``Mumford's theorem,'' that appeared in Segal's monograph [Seg88, Seg04], classifying complex extensions, also known as one-dimensional modular functors. Algebraically, the real extensions are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The real determinant line bundle encodes the trace anomaly of euclidean two-dimensional conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. It serves as the guiding example in our setup and proof strategy, providing the universal $\mathbb{R}_+$-extension. Our results thus give a mathematical explanation for the appearance of the central charge in CFT and SLE, and prove the conjectured uniqueness of Brownian loop measures as the canonical restriction function for SLE.

math-ph↗

arXiv2026

Multiple SLEs for $κ\in (0,8)$: Coulomb gas integrals and pure partition functions

In this article, we give an explicit relationship of SLE partition functions with Coulomb gas formalism of conformal field theory. We first construct a family of SLE$(κ)$ partition functions as Coulomb gas integrals and derive their various properties. In accordance with an interpretation as probabilistic correlations in loop $O(n)$ models, they are always positive when $κ\in (8/3,8)$, while they may have zeroes for $κ\le 8/3$. They also admit a Frobenius series expansion that matches with the algebraic content from CFT. Moreover, we check that at the first level of fusion, they have logarithmic asymptotic behavior when $κ= 8/3$ and $κ= 8$, in accordance with logarithmic minimal models $M(2,1)$ and $M(2,3)$, respectively. Second, we construct $\SLE_κ$ pure partition functions and show that they are real-analytic in $κ\in (0,8)$ and decay to zero as a polynomial of $(8-κ)$ as $κ\to 8$. We explicitly relate the Coulomb gas integrals and pure partition functions together in terms of the meander matrix. As a by-product, our results yield a construction of global non-simple multiple chordal SLE$(κ)$ measures ($κ\in (4,8)$) uniquely determined by their re-sampling property.

math-ph↗

arXiv2026

Large deviations of Dyson Brownian motion on the circle and multiradial SLE(0+)

We show a finite-time large deviation principle (LDP) for "Dyson type" diffusion processes, including Dyson Brownian motion (DBM) on the circle, for a fixed number of particles as the coupling parameter $β=8/κ$ tends to $+\infty$. Zero-energy systems correspond to the Calogero-Moser-Sutherland integrable system. We also characterize the large-time behavior of finite-energy systems: zero-energy systems approach exponentially fast a static equally-spaced configuration, while finite-energy systems may have polynomial convergence rates, and the system may never become static. We use our DBM result to derive a finite-time LDP in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$_κ$, as $κ$ tends to $0+$, with good rate function being the multiradial Loewner energy. Using a derivative estimate for the radial Loewner map in terms of the energy of its driving function, we show that finite-energy multiradial Loewner hulls are disjoint unions of simple curves, except at their common endpoint.

math.PR↗

arXiv2026

Multiradial SLE with spiral: resampling property and boundary perturbation

We consider multiple radial Schramm-Loewner evolution (SLE) curves with various time parameterizations and possible spiraling behavior. We construct them by tilting independent radial SLEs with a suitable local martingale, generalizing the earlier construction by Healey and Lawler. We prove that the curves are almost surely transient (i.e., they emanate from boundary points and terminate at a common interior target point). We show that they enjoy the resampling property: conditional on all of the curves but one, the remaining curve is distributed as chordal SLE in the remaining domain. We also verify that the multiradial SLE measure satisfies a natural boundary perturbation property analogous to that of the known SLE variants, involving its partition function (which is finite). Interestingly, in the parlance of Coulomb gas formalism in conformal field theory, partition functions of multiradial SLE processes with spiral involve both electric and magnetic charges.

math.PR↗

arXiv2026

On the geometry of locally growing Loewner chains

Loewner chains are ubiquitous in the theory of slit mappings, and hence in the study of bounded conformal maps. They have attracted new interest in the past decades through their applications to statistical physics and fractal geometry, particularly in contexts involving randomness. In this article, we delve into topological features of the growing hulls obtained from Loewner chains with a general local growth property, inspired by the classical works of Loewner and Pommerenke. We first revisit Loewner's theorem, associating to each locally growing collection of hulls a real-valued driving function W, possibly discontinuous. We then investigate the points chronologically added to the growing hulls, which may be part of a simply connected swallowed ``bubble'', or a compact connected boundary set. For continuous driving functions, the Loewner chain can often be associated with a continuous curve (dubbed ``generating curve''). Motivated by this, we introduce a more general notion of a ``generating function'' for the Loewner chain, and characterize when there exists such a function η (which can be continuous, càdlàg, càglàd, or neither). We then investigate the necessity of left and right limits for η from the point of view of the topology of the growing hulls. We find in particular that left-continuity implies path-connectedness and local connectedness of the hulls, as well as the existence of right limits, whereas failure of left-continuity leads to pathological boundary behavior.

math.CV↗

arXiv2026

Level lines of the Gaussian free field and $c=1$ degenerate conformal blocks

We consider Gaussian free field (GFF) on simply connected domains with piecewise constant Dirichlet boundary data. We show that the crossing probabilities for its level lines are determined by conformal blocks of primary fields in a conformal field theory (CFT) with central charge $c = 1$ which are degenerate at each insertion. Alternatively, the crossing probabilities are ratios of explicit partition functions of fused multiple $\mathrm{SLE}_4$ curves, which can be written in terms of fused Specht polynomials introduced recently by Lafay, Peltola, & Roussillon in a representation-theoretic context. We also prove that for the metric graph GFF introduced by Lupu, with appropriate boundary conditions, the crossing probabilities for its level sets converge in the scaling limit to our formulas. In particular, the geometry of the level-set percolation for both the continuum GFF and the metric graph GFF has a CFT description in terms of the aforementioned $c=1$ conformal blocks, which are linearly independent and solve the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs of arbitrary orders. Interestingly, not all combinatorial boundary conditions are amenable for the GFF models -- while the CFT contains conformal blocks of primary fields labeled by generalized Dyck paths (i.e., semi-standard Young tableaux), the ones appearing in the above models satisfy specific monotonicity constraints.

math-ph↗

arXiv2026

Irregular SLE(4) martingales and isomonodromic deformations

We consider non-Fuchsian monodromy preserving deformations on a Riemann sphere. The associated isomonodromic deformation parameters on this surface comprise the positions of the singularities, together with the Birkhoff (spectral) invariants owing to the presence of irregular singularities. Our first main result is the derivation of the Loewner evolution of these isomonodromic deformation parameters. Using this result, we construct martingale observables for Schramm-Loewner evolution (SLE(4)) processes in the presence of double poles. Geometrically, the expressions contain the pre-Schwarzian and Schwarzian of the Loewner evolution, arising from conformal covariance of the observable. Furthermore, we characterize these SLE(4) observables uniquely in terms of confluent BPZ equations of a CFT with central charge c=1.

math-ph↗

arXiv2026

Large deviations of SLE(0+) variants in the capacity parameterization

We prove large deviation principles (LDPs) for full chordal, radial, and multichordal SLE(0+) curves parameterized by capacity. The rate function is given by the appropriate variant of the Loewner energy. There are two key novelties in the present work. First, we strengthen the topology in the known chordal LDPs into the topology of full parameterized curves including all curve endpoints. We also obtain LDPs in the space of unparameterized curves. Second, we address the radial case, which requires in part different methods from the chordal case, due to the different topological setup. We establish our main results via proving an exponential tightness property and combining it with detailed curve escape probability estimates, in the spirit of exponentially good approximations in LDP theory. In the radial case, additional work is required to refine the estimates appearing in the literature. Notably, since we manage to prove a finite-time LDP in a better topology than in earlier literature, escape energy estimates follow as a consequence of the escape probability estimates.

math.PR↗

arXiv2026

Multiradial Schramm-Loewner evolution: Infinite-time large deviations and transience

In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(κ)$, as $κ\to 0$, with good rate function being the multiradial Loewner energy. Here, we extend this result to infinite time in the topology of common-capacity-parameterized curves, and streamline the proof. A main step is to derive detailed escape probability estimates for multiradial SLE$(κ)$ curves in the common parameterization, which extend the single-curve estimates achieved in [AP26]. As a by-product, we also get that multiradial SLE$(κ)$ curves, with $κ\leq 8/3$, are transient at their common terminal point, generalizing [FL15, HL21]. As a corollary to the LDP result, we obtain explicit asymptotics of the Brownian loop measure interaction term for finite-energy radial multichords, which is linear in the capacity-time and coincides with a certain choice of a cocycle for the Virasoro algebra.

math.PR↗

arXiv2025

Uniform spanning trees and random matrix statistics

We consider a uniform spanning tree in a $δ$-square grid approximation of a planar domain $Ω$. For given integer $n\ge 2$, we condition the tree on the following $n$-arm event: we pick $n$ branches, emanating from $n$ points microscopically close to a given interior point, and condition them to connect to the boundary $\partial Ω$ without intersecting. What can be said about the geometry of these branches? We derive an exact formula for the characteristic function of the total winding of the branches. A surprising consequence of this formula is that in the scaling limit, the behaviour of this function depends on the total number of branches $n$ only through its parity. We also describe the scaling limit of the branches. If $Ω$ is the unit disc, then they hit the boundary (i.e., the unit circle) at random positions which coincide exactly with the eigenvalues of a random matrix of size $n$ drawn from the Circular Orthogonal Ensemble (COE, also called C$β$E with $β=1$). Furthermore, the branches converge to Loewner evolution driven by the circular Dyson Brownian motion with parameter $β= 4$ (i.e., $n$-sided radial SLE$_κ$ with $κ=2$). We thus verify a prediction made by Cardy in this setting. Along the way, we develop a flow-line (imaginary geometry) coupling of $n$-sided radial SLE$_κ$ with the Gaussian free field, which may be of independent interest. Surprisingly, we find that the variance of the corresponding field near the singularity also does not depend on the number $n\ge 2$ of curves. In contrast, the variance of the the winding of the curves behaves as $κ/n^2$, which agrees with the predictions from the physics literature made by Wieland and Wilson numerically, and by Duplantier and Binder using Coulomb gas methods -- but disagrees with a result of Kenyon.

math.PR↗

arXiv2025

Loewner traces driven by Levy processes

Loewner chains with Levy drivers have been proposed as models for random dendritic growth in two dimensions, and as candidates for finding extremal multifractal spectra in problems in classical function theory. These processes are not scale-invariant in general, but they do enjoy a natural domain Markov property thanks to the stationary independent increments of Levy processes. The associated Loewner hulls feature remarkably intricate topological properties, of which very little is known rigorously. We prove that a chordal Loewner chain driven by a Levy process $W$ satisfying mild regularity conditions (including stable processes) is a.s. generated by a cadlag curve. Specifically, if the diffusivity parameter of the driving process $W$ is $κ\in [0,8)$, then the jump measure of $W$ is required to be locally (upper) Ahlfors regular near the origin, while if $κ> 8$, no constraints are imposed. In particular, we show that the associated Loewner hulls are a.s. locally connected and path-connected. We also show that, the complements of the hulls are a.s. Holder domains when $κ\neq 4$ (which is not expected to hold when $κ=4$), without any regularity assumptions. The proofs of these results mainly rely on careful derivative estimates for both the forward and backward Loewner maps obtained using delicate but robust enough supermartingale domination arguments. As one cannot control the jump accumulation of general Levy processes, we must circumvent all reasoning that would use continuity. To prove the local connectedness, we use an extension of part of the Hahn-Mazurkiewicz theorem: hulls generated by cadlag curves are locally connected even when jumps would occur at infinite intensity.

math.PR↗

arXiv2025

Planar UST Branches and $c=-2$ Degenerate Boundary Correlations

We provide a conformal field theory (CFT) description of the probabilistic model of boundary effects in the wired uniform spanning tree (UST) and its algebraic content, concerning the entire first row of the Kac table with central charge $c=-2$. Namely, we prove that all boundary-to-boundary connection probabilities for (potentially fused) branches in the wired UST converge in the scaling limit to explicit CFT quantities, expressed in terms of determinants, which can also be viewed as conformal blocks of degenerate primary fields in a boundary CFT with central charge $c=-2$. Moreover, we verify that the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs (i.e., Virasoro degeneracies) of arbitrary orders hold, and we also reveal an underlying valenced Temperley-Lieb algebra action on the space of boundary correlation functions of primary fields in this model. To prove these results, we combine probabilistic techniques with representation theory.

math-ph↗

arXiv2025

Fused Specht Polynomials and $c=1$ Degenerate Conformal Blocks

We introduce a class of polynomials that we call fused Specht polynomials and use them to characterize irreducible representations of the fused Hecke algebra with parameter $q=-1$ in the space of polynomials. We apply the fused Specht polynomials to construct a basis for a space of holomorphic (chiral) conformal blocks with central charge $c=1$ which are degenerate at each point. In conformal field theory, this corresponds to all primary fields having conformal weight in the Kac table. The associated correlation functions are expected to give rise to conformally invariant boundary conditions for the Gaussian free field, which has also been verified in special cases.

math-ph↗

arXiv2025

From the Conformal Anomaly to the Virasoro Algebra

The conformal anomaly and the Virasoro algebra are fundamental aspects of 2D conformal field theory and conformally covariant models in planar random geometry. In this article, we explicitly derive the Virasoro algebra from an axiomatization of the conformal anomaly in terms of real determinant lines, one-dimensional vector spaces associated to Riemann surfaces with analytically parametrized boundary components. Here, analytical orientation-preserving diffeomorphisms and deformations of the circle naturally act on the boundary components. We introduce a sewing operation on the real determinant lines over the semigroup of annuli, which then induces central extensions of the diffeomorphism group, as well as of the complex deformations. Our main theorem shows that on the one hand, the cocycle associated to the central extension of diffeomorphisms is trivial, while on the other hand, the Lie algebra cocycle associated to the central extension of complex deformations is nontrivial, yielding the imaginary part of the Gel'fand-Fuks cocycle. We thus answer a question, partly negatively and partly affirmatively, discussed by Andre Henriques and Dylan Thurston in 2011. The proof uses concrete computations, which we aim to be accessible to a wide audience. We also show an explicit relation to loop Loewner energy, anticipating the real determinant lines to be pertinent to locally conformally covariant (Malliavin-Kontsevich-Suhov) measures on curves and loops, as well as to Kähler geometry and geometric quantization of moduli spaces of Riemann surfaces. Inherently, the conformal anomaly and real determinant line bundles are expected to be universal, following a classification of modular functors.

math-ph↗

arXiv2024

On the spin interface distribution for non-integrable variants of the two-dimensional Ising model

We point out that the construction of a martingale observable describing the spin interface of the two-dimensional Ising model extends to a class of non-integrable variants of the two-dimensional Ising model, and express it in terms of Grassmann integrals. Under a conjecture about the scaling limit of this object, which is similar to some results recently obtained using constructive renormalization group methods, this would imply that the distribution of the interface at criticality has the same scaling limit as in the integrable model: Schramm-Loewner evolution SLE(3).

math-ph↗

arXiv2024

Connection Probabilities of Multiple FK-Ising Interfaces

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight $q \in [1,4)$. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of $q$ than the FK-Ising model ($q=2$). Given the convergence of interfaces, the conjectural formulas for other values of $q$ could be verified similarly with relatively minor technical work. The limit interfaces are variants of $\mathrm{SLE}_κ$ curves (with $κ= 16/3$ for $q=2$). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all $q \in [1,4)$, thus providing further evidence of the expected CFT description of these models.

math.PR↗

arXiv2024

Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT

We find explicit SLE(8) partition functions for the scaling limits of Peano curves in the uniform spanning tree (UST) in topological polygons with general boundary conditions. They are given in terms of Coulomb gas integral formulas, which can also be expressed in terms of determinants involving a-periods of a hyperelliptic Riemann surface. We also identify the crossing probabilities for the UST Peano curves as ratios of these partition functions. The partition functions are interpreted as correlation functions in a logarithmic conformal field theory (log-CFT) of central charge $c= -2$. Indeed, it is clear from our results that this theory is not a minimal model and exhibits logarithmic phenomena -- the limit functions have logarithmic asymptotic behavior, that we calculate explicitly. General fusion rules for them could also be inferred from the explicit formulas. The discovered algebraic structure matches the known Virasoro staggered module classification, so in this sense, we give a direct probabilistic construction for correlation functions in a log-CFT of central charge $-2$ describing the UST model.

math.PR↗