arXiv subjects

Roland M. Friedrich

Publications and source records attributed to Roland M. Friedrich.

15 recordsLinked to original sources

arXiv2025

The geometry of moral decision making

We show how (resource) bounded rationality can be understood as the interplay of two fundamental moral principles: deontology and utilitarianism. In particular, we interpret deontology as a regularisation function in an optimal control problem, coupled with a free parameter, the inverse temperature, to shield the individual from expected utility. We discuss the information geometry of bounded rationality and aspects of its relation to rate distortion theory. A central role is played by Markov kernels and regular conditional probability, which are also studied geometrically. A gradient equation is used to determine the utility expansion path. Finally, the framework is applied to the analysis of a disutility model of the restriction of constitutional rights that we derive from legal doctrine. The methods discussed here are also relevant to the theory of autonomous agents.

cs.IT↗

arXiv2018

Types, Codes and TFTs

We establish a relation between fully extended $2$-dimensional TQFTs and recognisable weighted formal languages, rational biprefix codes and lattice TFTs. We show the equivalence of $2D$ closed TFTs and rational exchangeable series and we discuss the important special case of finite groups. Finally, we outline a reformulation in terms of a restricted version of second order monadic logic.

math.RA↗

arXiv2018

Generalised Operations in Free Harmonic Analysis

This article, which is substantially motivated by the previous joint work with J. McKay [8], establishes the analytic analogues of the relations we found free probability has with Witt vectors. Therefore, we first present a novel analytic derivation of an exponential map which relates the free additive convolution on $\mathbb{R}$ with the free multiplicative convolution on either the unit circle or $\mathbb{R}_+^*$, for compactly supported, freely infinitely divisible probability measures. We then introduce several new operations on these measures, which gives rise to more extended classes of operations. Then we consider the relation with classical infinite divisibility and the geometry of the spaces involved. Finally, we discuss the general structure, using the language of operads and algebraic theories, of the various operations we defined give rise to.

math.PR↗

arXiv2017

(Co)monads in Free Probability Theory

We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free probability is isomorphic to several well-known categories of algebras, such as, e.g., Witt vectors, differential algebras, etc. Within this framework moment-cumulant formulae are shown to correspond to natural transformations and not to be exclusive to probability theories. Finally, we start to discuss free probability and in particular free harmonic analysis from the point of view of algebraic theories and operads.

math.PR↗

arXiv2015

Homogeneous Lie Groups and Quantum Probability

Here we extend the algebro-geometric approach to free probability, started in~\cite{FMcK4,F14}, to general (non)-commutative probability theories. We show that any universal convolution product of moments of independent (non)-commutative random variables defined on a graded connected dual semi-group is given by a pro-unipotent group scheme. We show that moment-cumulant formulae have a natural interpretation within the theory of homogeneous Lie groups, which we generalise for the present purpose, and are given by the log and exp map, respectively. Finally, we briefly discuss the universal role of the shuffle Hopf algebra.

math.RT↗

arXiv2013

Almost Commutative Probability Theory

We solve two longstanding major problems in Free Probability. This is achieved by generalising the theory to one with values in arbitrary commutative algebras. We prove the existence of the multi-variable $S$-transform, and show that it is naturally realised as a faithful linear representation. Further, we prove that in dimension one, the analog of the classical relation between addition and multiplication of independent random variables holds for free random variables, if the co-domain is an algebra over the rationals. In this case the multiplicative problem can be reduced to the additive one, which is not true in dimensions greater than one. Finally, we classify the groups which arise as joint distributions of $n$-tuples of non-commutative random variables, endowed with the free convolution product, which is the binary operation that encodes the multiplication of free $n$-tuples.

math.PR↗

arXiv2009

The Global Geometry of Stochastic Lœwner Evolutions

In this article we develop a concise description of the global geometry which is underlying the universal construction of all possible generalised Stochastic Lœwner Evolutions. The main ingredient is the Universal Grassmannian of Sato-Segal-Wilson. We illustrate the situation in the case of univalent functions defined on the unit disc and the classical Schramm-Lœwner stochastic differential equation. In particular we show how the Virasoro algebra acts on probability measures. This approach provides the natural connection with Conformal Field Theory and Integrable Systems.

math-ph↗

arXiv2008

A Renormalisation Group approach to Stochastic Lœwner Evolutions and the Doob h-transform

In this notes we shall describe the relation of a certain class of simple random curves arising in 2D statistical mechanics models in the scaling limit, which can be described dynamically by Stochastic Lœwner Evolutions (SLE), and the equivalent Renormalisation Group (RG) theoretic interpretation in Conformal Field Theory, as a fixed point of the RG flow. Further, we shall recall the relation of this random curves with String Theory, and how one can derive a general measure on such random paths, by using weighted regularised determinants, which come from sections of twisted line bundles. Importantly, the null vector at level two in the Verma module for the highest-weight representation of the Virasoro algebra corresponds to a generalised Doob-Getoor h-transform.

hep-th↗

arXiv2006

The Hilbert-uniformization is real-analytic

In \cite{Boed}, C.-F. Bödigheimer constructed a finite cell-complex $\mf{Par}_{g,n,m}$ and a bijective map $\cH: \mf{Dip}_{g,n,m} \to \mf{Par}_{g,n,m}$ (the Hilbert-uniformization) from the moduli space of dipole functions on Riemann surfaces with $n$ directions and $m$ punctures to $\mf{Par}_{g,n,m}$. In \cite{Boed} and \cite{Eb}, it is proven that $\cH$ is a homeomorphism. The first result of this note is that the space $\mf{Dip}_{g,n,m}$ carries a natural structure of a real-analytic manifold. Our second result is that $\cH$ is real-analytic, at least on the preimage of the top-dimensional open cells of $\mf{Par}_{g,n,m}$.

math.DG↗

arXiv2005

The Correlator Toolbox, Metrics and Moduli

We discuss the possible set of operators from various boundary conformal field theories to build meaningful correlators that lead via a Loewner type procedure to generalisations of SLE($κ,ρ$). We also highlight the necessity of moduli for a consistent kinematic description of these more general stochastic processes. As an illustration we give a geometric derivation of $\text{SLE}(κ,ρ)$ in terms of conformally invariant random growing compact subsets of polygons. The parameters $ρ_j$ are related to the exterior angles of the polygons. We also show that $\text{SLE}(κ,ρ)$ can be generated by a Brownian motion in a gravitational background, where the metric and the Brownian motion are coupled. The metric is obtained as the pull-back of the Euclidean metric of a fluctuating polygon.

hep-th↗

arXiv2005

Diffusing polygons and SLE($κ,ρ$)

We give a geometric derivation of SLE($κ,ρ$) in terms of conformally invariant random growing subsets of polygons. We relate the parameters $ρ_j$ to the exterior angles of the polygons. We also show that SLE($κ,ρ$) can be generated by a metric Brownian motion, where metric and Brownian motion are coupled and the metric ist the pull-back metric of the Euclidean metric of an evolving polygon.

math.PR↗

arXiv2005

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that "consistent," conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter $κ$) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if $κ=6$.

math.PR↗

arXiv2005

On Chordal and Bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner's equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by a real constant and a function homogeneous of degree minus one which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property if the interaction term vanishes and the real parameter equals six.

math.PR↗

arXiv2004

Stochastic Loewner evolution in multiply connected domains

We construct radial stochastic Loewner evolution in multiply connected domains, choosing the unit disk with concentric circular slits as a family of standard domains. The natural driving function or input is a diffusion on the associated Teichmüller space. The diffusion stops when it reaches the boundary of the Teichmüller space. We show that for this driving function the family of random growing compacts has a phase transition for $κ=4$ and $κ=8$, and that it satisfies locality for $κ=6$.

math.PR↗

arXiv2000

Finite-Volume Fractional-Moment Criteria for Anderson Localization

A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the ``multiscale analysis''. In the converse direction, the analysis rules out fast power-law decay of the Green functions at mobility edges.

math-ph↗