arXiv subjects

Stephan Gufler

Publications and source records attributed to Stephan Gufler.

10 recordsLinked to original sources

arXiv2024

On support sets of the critical Liouville Quantum Gravity

We study fractal properties of support sets of the critical Liouville Quantum Gravity (cLQG) associated with the Gaussian Free Field in planar domains. Specifically, we completely characterize the gauge functions $\phi$ (subject to mild monotonicity conditions) for which the cLQG admits a support set of finite $\phi$-Hausdorff measure. As a corollary, we settle the conjecture that the cLQG is supported on a set of vanishing Hausdorff dimension. Our proofs are based on the fact that the cLQG describes the near-critical level sets of Discrete Gaussian Free Field.

math.PR↗

arXiv2023

AMP algorithms and Stein's method: Understanding TAP equations with a new method

We propose a new iterative construction of solutions of the classical TAP equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager correction. The algorithm can be started in an arbitrary point, and converges up to the AT line. The analysis relies on a novel treatment of mean field algorithms through Stein's method. As such, the approach also yields weak convergence of the effective fields at all temperatures towards Gaussians, and can be applied, upon proper alterations, to all models where TAP-like equations and a Stein-operator are available.

math.PR↗

arXiv2023

On concavity of TAP free energy in the SK model

We analyse the Hessian of the Thouless-Anderson-Palmer (TAP) free energy for the Sherrington-Kirkpatrick model, below the de Almeida-Thouless line, evaluated in Bolthausen's approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, an that the support includes zero on the AT line. In this ``macroscopic'' sense, TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetisations. This proves a spectral interpretation of the AT line. However, for specific magnetizations, the Hessian of the TAP free energy can have positive outlier eigenvalues. The question whether such outliers may also occur close to the TAP solutions is left open. In a simplified setting where the magnetizations are independent of the disorder, we prove that Plefka's second condition is equivalent to all eigenvalues being negative.

math.PR↗

arXiv2022

Evolving genealogies for branching populations under selection and competition

For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by Donnelly and Kurtz (1999), namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of ``potential'' selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the ``active'' selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz' Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation.

math.PR↗

arXiv2021

TAP equations are repulsive

We show that for low enough temperatures, but still above the AT line, the Jacobian of the TAP equations for the SK model has a macroscopic fraction of eigenvalues outside the unit interval. This provides a simple explanation for the numerical instability of the fixed points, which thus occurs already in high temperature. The insight leads to some algorithmic considerations on the low temperature regime, also briefly discussed.

math.PR↗

arXiv2020

Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions stays negative as the ratio of the radii of the inner and the outer boundary tends to infinity.

math.PR↗

arXiv2020

Near-maxima of the two-dimensional Discrete Gaussian Free Field

We consider the Discrete Gaussian Free Field (DGFF) in domains $D_N\subseteq\mathbb Z^2$ arising, via scaling by $N$, from nice domains $D\subseteq\mathbb R^2$. We study the statistics of the values order $\sqrt{\log N}$ below the absolute maximum. Encoded as a point process on $D\times\mathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the absolute maximum) tends, as $N\to\infty$, in law to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence holds jointly with the extremal process, for which $Z^D$ enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with $Z^D$. While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to "norm" the lattice cLQG measure lead to the same limit object, modulo overall normalization.

math.PR↗

arXiv2017

Pathwise construction of tree-valued Fleming-Viot processes

In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued $Ξ$-Fleming-Viot processes and deduce path properties. For instance, these processes usually have a.s. càdlàg paths with jumps at the times of large reproduction events. In the case of coming down from infinity, the construction on the lookdown space also allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a.s. càdlàg paths with additional jumps at the extinction times of parts of the population.

math.PR↗

arXiv2017

A representation for exchangeable coalescent trees and generalized tree-valued Fleming-Viot processes

We give a de Finetti type representation for exchangeable random coalescent trees (formally described as semi-ultrametrics) in terms of sampling iid sequences from marked metric measure spaces. We apply this representation to define versions of tree-valued Fleming-Viot processes from a $Ξ$-lookdown model. As state spaces for these processes, we use, besides the space of isomorphy classes of metric measure spaces, also the space of isomorphy classes of marked metric measure spaces and a space of distance matrix distributions. This allows to include the case with dust in which the genealogical trees have isolated leaves.

math.PR↗

arXiv2017

Invariance principles for tree-valued Cannings chains

We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under the conditions of Möhle and Sagitov, this convergence holds for all tree-valued Fleming-Viot processes under consideration in the dust-free case, and for the Fleming-Viot processes with values in the space of distance matrix distributions in the case with dust. Convergence to Fleming-Viot processes with values in the space of marked metric measure spaces in the case with dust is ensured by an additional assumption on the probability that a randomly sampled individual belongs to a non-singleton family.

math.PR↗