arXiv subjects

Christof Schmidhuber

Publications and source records attributed to Christof Schmidhuber.

At least 19 recordsLinked to original sources

arXiv2025

Critical Dynamics of Random Surfaces and Multifractal Scaling

The critical dynamics of conformal field theories on random surfaces is investigated beyond the previously studied dynamics of the overall area and the genus. It is found that the evolution of the order parameter in physical time performs a generalization of the multifractal random walk. Accordingly, the higher moments of time variations of the order parameter exhibit multifractal scaling. The series of Hurst exponents is computed and illustrated at the examples of the Ising-, 3-state-Potts-, and general minimal models as well as $c=1$ models on a random surface. It is noted that some of these models can replicate the observed multifractal scaling in financial markets.

hep-th↗

arXiv2025

Critical Dynamics of Random Surfaces: Time Evolution of Area and Genus

Conformal field theories with central charge $c\le1$ on random surfaces have been extensively studied in the past. Here, this discussion is extended from their equilibrium distribution to their critical dynamics. This is motivated by the conjecture that these models describe the time evolution of certain social networks that are self-driven to a critical point. This paper focuses on the dynamics of the overall area and the genus of the surface. The time evolution of the area is shown to follow a Cox Ingersol Ross process. Planar surfaces shrink, while higher genus surfaces grow to a size of order of the inverse cosmological constant. The time evolution of the genus is argued to lead to two different phases, dominated by (i) planar surfaces, and (ii) ``foamy'' surfaces, whose genus diverges. In phase (i), which exhibits critical phenomena, time variations of the order parameter are approximately t-distributed with 4 or more degrees of freedom.

hep-th↗

arXiv2025

Trends and Reversion in Financial Markets on Time Scales from Minutes to Decades

We empirically analyze the reversion of financial market trends with time horizons ranging from minutes to decades. The analysis covers equities, interest rates, currencies and commodities and combines 14 years of futures tick data, 30 years of daily futures prices, 330 years of monthly asset prices, and yearly financial data since medieval times. Across asset classes, we find that markets are in a trending regime on time scales that range from a few hours to a few years, while they are in a reversion regime on shorter and longer time scales. In the trending regime, weak trends tend to persist, which can be explained by herding behavior of investors. However, in this regime trends tend to revert before they become strong enough to be statistically significant, which can be interpreted as a return of asset prices to their intrinsic value. In the reversion regime, we find the opposite pattern: weak trends tend to revert, while those trends that become statistically significant tend to persist. Our results provide a set of empirical tests of theoretical models of financial markets. We interpret them in the light of a recently proposed lattice gas model, where the lattice represents the social network of traders, the gas molecules represent the shares of financial assets, and efficient markets correspond to the critical point. If this model is accurate, the lattice gas must be near this critical point on time scales from 1 hour to a few days, with a correlation time of a few years.

q-fin.ST↗

arXiv2021

Financial Markets and the Phase Transition between Water and Steam

Motivated by empirical observations on the interplay of trends and reversion, a lattice gas model of financial markets is presented. The shares of an asset are modeled by gas molecules that are distributed across a hidden social network of investors. The model is equivalent to the Ising model on this network, whose magnetization represents the deviation of the asset price from its value. Moreover, the system should drive itself to its critical temperature in efficient markets. There, it is characterized by universal critical exponents, in analogy with the second-order phase transition between water and steam. These critical exponents imply predictions for the auto-correlations of financial market returns and for Hurst exponents. For a simple network topology, consistency with empirical observations implies a fractal network dimension near 3, and a correlation time at least as long as the economic cyle. To also explain the observed market auto-correlations at intermediate scales, the model should be extended beyond the critical domain, to other network topologies, and to other models of critical dynamics.

q-fin.ST↗

arXiv2021

Chaitin's Omega and an Algorithmic Phase Transition

We consider the statistical mechanical ensemble of bit string histories that are computed by a universal Turing machine. The role of the energy is played by the program size. We show that this ensemble has a first-order phase transition at a critical temperature, at which the partition function equals Chaitin's halting probability $Ω$. This phase transition has curious properties: the free energy is continuous near the critical temperature, but almost jumps: it converges more slowly to its finite critical value than any computable function. At the critical temperature, the average size of the bit strings diverges. We define a non-universal Turing machine that approximates this behavior of the partition function in a computable way by a super-logarithmic singularity, and discuss its thermodynamic properties. We also discuss analogies and differences between Chaitin's Omega and the partition functions of a quantum mechanical particle, a spin model, random surfaces, and quantum Turing machines. For universal Turing machines, we conjecture that the ensemble of bit string histories at the critical temperature has a continuum formulation in terms of a string theory.

cs.CC↗

arXiv2020

Trends, Reversion, and Critical Phenomena in Financial Markets

Financial markets across all asset classes are known to exhibit trends. These trends have been exploited by traders for decades. Here, we empirically measure when trends revert, based on 30 years of daily futures prices for equity indices, interest rates, currencies and commodities. We find that trends tend to revert once they reach a critical level of statistical significance. Based on polynomial regression, we carefully measure this critical level. We find that it is universal across asset classes and has a universal scaling behavior, as the trend's time horizon runs from a few days to several years. The corresponding regression coefficients are small, but statistically highly significant, as confirmed by bootstrapping and out-of-sample testing. Our results signal to investors when to exit a trend. They also reveal how markets have become more efficient over the decades. Moreover, they point towards a potential deep analogy between financial markets and critical phenomena: our analysis supports the conjecture that financial markets can be modeled as statistical mechanical ensembles of Buy/Sell orders near critical points. In this analogy, the trend strength plays the role of an order parameter, whose dynamcis is described by a Langevin equation with a quartic potential.

q-fin.ST↗

arXiv2002

Old Puzzles

In this talk I first briefly review how far we have come in answering old questions about the most fundamental building blocks of matter. I begin with things we know, which is the Standard Model, and then talk about things we can guess, which is superstring theory. After this review I discuss a key point at which our understanding of superstring theory presently stops: the problem of supersymmetry breaking and the cosmological constant. I explain in which direction I imagine a way out. This way out predicts gravitinos and dilatons with masses of order milli-eV.

hep-th↗

arXiv2001

Brane Supersymmetry Breaking and the Cosmological Constant: Open Problems

It has recently been argued that non--BPS brane world scenarios can reproduce the small value of the cosmological constant that seems to have been measured. Objections against this proposal are discussed and necessary (but not sufficient) conditions are stated under which it may work. At least n=2 extra dimensions are needed. Also, the mass matrix in the supergravity sector must satisfy $Str M^2=0$. Moreover, the proposal can be ruled out experimentally if Newton's constant remains unchanged down to scales of 10 micrometers. If, on the other hand, such a ``running Newton constant'' is observed, it could provide crucial experimental input for superstring phenomenology.

hep-th↗

arXiv2000

Strings from Logic

What are strings made of? The possibility is discussed that strings are purely mathematical objects, made of logical axioms. More precisely, proofs in simple logical calculi are represented by graphs that can be interpreted as the Feynman diagrams of certain large-N field theories. Each vertex represents an axiom. Strings arise, because these large-N theories are dual to string theories. These ``logical quantum field theories'' map theorems into the space of functions of two parameters: N and the coupling constant. Undecidable theorems might be related to nonperturbative field theory effects.

hep-th↗

arXiv2000

Micrometer Gravitinos and the Cosmological Constant

We compute the 4--dimensional cosmological constant in string compactifications in which the Standard Model fields live on a non-supersymmetric brane inside a supersymmetric bulk. The cosmological constant receives contributions only from the vacuum energy of the bulk supergravity fields, but not from the vacuum energy of the brane fields. The latter is absorbed in a warp factor. Supersymmetry breaking on the brane at the TeV scale implies supersymmetry breaking in the bulk at the micrometer scale. This produces a tiny cosmological constant that agrees with experiment within a few orders of magnitude. Our argument predicts superpartners of the graviton with mass of order $10^{-3}$ eV. They should be observable in short-distance tests of Einstein Gravity.

hep-th↗

arXiv2000

AdS_5 and the 4D Cosmological Constant

The hypothesis is discussed that our universe is really 5--dimensional with a nonzero cosmological constant that produces a large negative curvature. In this scenario, the observable flat 4--dimensional universe is identified with the holographic projection of the 5--dimensional world onto its own boundary.

hep-th↗

arXiv2000

AdS-Flows and Weyl Gravity

An analogy is noted between RG flow equations in 4-dimensional gauge theory, as derived from the AdS/CFT correspondence, and the RG flow equations in 4-dimensional field theory coupled to a particular limit of Weyl supergravity. This suggests a possible theory of dynamical 3-branes with fluctuating 4-dimensional conformal factor. The argument involves a map from flows in 4-dimensional gauge theories to flows in a class of 2-dimensional sigma models.

hep-th↗

arXiv2000

Strings and Branes in Nonabelian Gauge Theory

It is an old speculation that SU(N) gauge theory can alternatively be formulated as a string theory. Recently this subject has been revived, in the wake of the discovery of D-branes. In particular, it has been argued that at least some conformally invariant cousins of the theory have such a string representation. This is a pedagogical introduction to these developments for non-string theorists. Some of the existing arguments are simplified.

hep-lat↗

arXiv1998

Wrapped Branes and Confined Momentum

We present string-like soliton solutions of three-dimensional gravity, coupled to a compact scalar field $x^{11}$ and Kaluza-Klein reduced on a circle. These solitons carry fractional magnetic flux with respect to the Kaluza-Klein gauge field. Summing over such ``Kaluza-Klein flux tubes'' is shown to imply summing over a subclass of three-dimensional topologies (Seifert manifolds). It is also shown to imply an area law for the Wilson loop of the Kaluza-Klein gauge field; the confined charge is nothing but Kaluza-Klein momentum. Applied to the membrane of M-theory, this is interpreted as ``dynamical wrapping'' of the M-brane around its eleventh embedding dimension $x^{11}$.

hep-th↗

arXiv1997

Wrapped M-branes and Three-dimensional Topologies

The three-dimensional topologies of the membrane of M-theory can be constructed by performing Dehn surgery along knot lines. We investigate membranes wrapped around a circle and the correponding subset of topologies (Seifert manifolds). The knot lines are interpreted as magnetic flux tubes in an XY model coupled to Maxwell theory. In this model the eleventh dimension of M-theory gets ``eaten'' by the world-brane metric. There is argued to be a second-order phase transition at a critical value of the string coupling constant. The topology fluctuations that correspond to the knot lines are irrelevant in one phase while they condense in the other phase.

hep-th↗

arXiv1997

On Water, Steam and String Theory

This is a colloquium-style review lecture for physicists and non-physicists, as part of the requirements for ``Habilitation'' at the university of Bern: At a pressure of 220 atm. and a temperature of 374 Celsius there is a second-order phase transition between water and steam. Understanding it requires the concept of the renormalization group. Images from computer simulations of the lattice gas model (included) are used to explain its basic ideas. It is briefly reviewed how the renormalization group is used to compute critical coefficients for the water-steam phase transition, in good agreement with experiment. Applications in particle physics and string theory are mentioned. The appendix contains a sample of the author's results on renormalization group flows in theories with dynamical gravity and their relation to perturbative string theory: gravity modifies critical coefficients and phase diagrams, in agreement with numerical calculations, and leads to curious phenomena such as oscillating flows and quantum mechanical flows.

hep-th↗

arXiv1996

D-brane actions

Effective world-brane actions for solitons of ten-dimensional type IIA and IIB superstring theory are derived using the formulation of solitons as Dirichlet branes. The one-brane actions are used to recover predictions of SL(2,Z) strong-weak coupling duality. The two-brane action, which contains a hidden eleventh target space coordinate, is shown to be the eleven-dimensional supermembrane action. It can be thought of as the membrane action of `M-theory'.

hep-th↗

arXiv1995

RG flow on random surfaces with handles and closed string field theory

The renormalization group flow in two-dimensional field theories that are coupled to gravity has unusual features: First, the flow equations are second order in derivatives. Second, in the presence of handles the flow has quantum mechanical properties. Third, the beta functions contain the elementary higher-genus vertices of closed string field theory. This is demonstrated at simple examples and is applied to derive various results about gravitationally dressed beta functions. The possibility of interpreting closed string field theory as the theory of the renormalization group on random surfaces with random topology is considered.

hep-th↗