Canonical Measure Research Articles

Quantitative logarithmic equidistribution of the crucial measures

Let K be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let $$\phi \in K(z)$$ with $$\deg (\phi )\ge 2$$ . Recently, Rumely introduced a family of discrete probability measures $$\{\nu _{\phi ^n}\}$$ on the Berkovich line $$\mathbf{P }^1_{\text {K}}$$ over K which carry information about the reduction of conjugates of $$\phi $$ . In a previous article, the author showed that the measures $$\nu _{\phi ^n}$$ converge weakly to the canonical measure $$\mu _\phi $$ . In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of $$\mathbf{P }^1_{\text {K}}$$ . These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to $$\nu _{\phi ^n}$$ converge to the potential function attached to $$\mu _\phi $$ , as well as an approximation result for the Lyapunov exponent of $$\phi $$ .

Universality for the random-cluster model on isoradial graphs

We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1 \leq q \leq 4$ and discontinuous for $q > 4$. For $1 \leq q \leq 4$, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in $1+1$ dimensions.

Can Typicality Arguments Dissolve Cosmology’s Flatness Problem?

Several physicists, among them Hawking, Page, Coule, and Carroll, have argued against the probabilistic intuitions underlying fine-tuning arguments in cosmology and instead propose that the canonical measure on the phase space of Friedman-Robertson-Walker space-times should be used to evaluate fine-tuning. They claim that flat space-times in this set are actually typical on this natural measure and that therefore the flatness problem is illusory. I argue that they misinterpret typicality in this phase space and, moreover, that no conclusion can be drawn at all about the flatness problem by using the canonical measure alone.

Dynamic principle for ensemble control tools.

Dynamical equations describing physical systems in contact with a thermal bath are commonly extended by mathematical tools called "thermostats." These tools are designed for sampling ensembles in statistical mechanics. Here we propose a dynamic principle underlying a range of thermostats which is derived using fundamental laws of statistical physics and ensures invariance of the canonical measure. The principle covers both stochastic and deterministic thermostat schemes. Our method has a clear advantage over a range of proposed and widely used thermostat schemes that are based on formal mathematical reasoning. Following the derivation of the proposed principle, we show its generality and illustrate its applications including design of temperature control tools that differ from the Nosé-Hoover-Langevin scheme.

μ-term hybrid inflation and split supersymmetry

We consider μ-term hybrid inflation which, in its minimal format with gravity mediated supersymmetry breaking, leads to split supersymmetry. The MSSM μ-term in this framework is larger than the gravitino mass mG, and successful inflation requires mG (and hence also |μ|) ≳5×107 GeV, such that the gravitino decays before the LSP neutralino freezes out. Assuming universal scalar masses of the same order as mG, this leads to split supersymmetry. The LSP wino with mass ≃ 2 TeV is a plausible dark matter candidate, the gluino may be accessible at the LHC, and the MSSM parameter tan⁡β≃1.7 in order to be compatible with the measured Higgs boson mass. The tensor-to-scalar ratio r, a canonical measure of gravity waves, can be as high as 0.001.

Direct determination approach for the multifractal detrending moving average analysis.

In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.

μ -hybrid inflation with low reheat temperature and observable gravity waves

In $\mu$-hybrid inflation a nonzero inflaton vacuum expectation value induced by supersymmetry breaking is proportional to the gravitino mass $m_{3/2}$, which can be exploited to resolve the minimal supersymmetric standard model $\mu$ problem. We show how this scenario can be successfully implemented with $m_{3/2} \sim 1-100$ TeV and reheat temperature as low as $10^6$ GeV by employing a minimal renormalizable superpotential coupled with a well defined non-minimal K\"ahler potential. The tensor-to-scalar ratio $r$, a canonical measure of primordial gravity waves in most cases is less than or of the order of $10^{-6}-10^{-3}$.

Novelty induces behavioural and glucocorticoid responses in a songbird artificially selected for divergent personalities

Stress physiology is thought to contribute to individual differences in behaviour. In part this reflects the fact that canonical personality measures consist of responses to challenges, including novel objects and environments. Exposure to novelty is typically assumed to induce a moderate increase in glucocorticoids (CORT), although this has rarely been tested. We tested this assumption using great tits, Parus major, selected for divergent personalities (bold-fast and shy-slow explorers), predicting that the shy birds would exhibit higher CORT following exposure to a novel object. We also scored behavioural responses to the novel object, predicting that bold birds would more frequently approach the novel object and exhibit more abnormal repetitive behaviours. We found that the presence of a novel object did induce a moderate CORT response, but selection lines did not differ in the magnitude of this response. Furthermore, although both selection lines showed a robust CORT elevation to a subsequent restraint stressor, the CORT response was stronger in bold birds and this effect was specific to novel object exposure. Shy birds showed a strong positive phenotypic correlation between CORT concentrations following the novel object exposure and the subsequent restraint stress. Behaviourally, the selection lines differed in their response during novel object exposure: as predicted, bold birds more frequently approached the novel object and shy birds more strongly decreased overall locomotion during the novel object trial, but birds from both selection lines showed significant and similar frequencies of abnormal repetitive behaviours during novel object exposure. Our findings support the hypothesis that personality emerges as a result of correlated selection on behaviour and underlying endocrine mechanisms and suggest that the relationship between endocrine stress physiology and personality is context dependent.

Value distribution of the sequences of the derivatives of iterated polynomials

We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in $\mathbb{C}\setminus\{0\}$ under the derivatives of the iterations of a polynomials $f\in\mathbb{C}[z]$ of degree more than one towards the $f$-equilibrium (or canonical) measure $\mu_f$ on $\mathbb{P}^1$. We also show that for every $C^2$ test function on $\mathbb{P}^1$, the convergence is exponentially fast up to a polar subset of exceptional values in $\mathbb{C}$. A parameter space analog of the latter quantitative result for the monic and centered unicritical polynomials family is also established.

Equidistribution of the crucial measures in non-Archimedean dynamics

TextLet K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ∈K(z) with deg⁡(ϕ)≥2. In this paper we consider the family of functions ordResϕn(x), which measure the resultant of ϕn at points x in PK1, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function gμϕ(x,x) attached to the canonical measure of ϕ. Following this, we are able to prove an equidistribution result for Rumely's crucial measures νϕn, each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ. VideoFor a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00.

Axion, μ term, and supersymmetric hybrid inflation

We show how successful supersymmetric hybrid inflation is realized in realistic models where the resolution of the minimal supersymmetric standard model mu problem is intimately linked with axion physics. The scalar fields that accompany the axion, such as the saxion, are closely monitored during and after inflation to ensure that the axion isocurvature perturbations lie below the observational limits. The scalar spectral index n_s is about 0.96 - 0.97, while the tensor-to-scalar ratio r, a canonical measure of gravity waves, lies well below the observable range in our example. The axion domain walls are inflated away, and depending on the axion decay constant f_a and the magnitude of the mu parameter, the axions and/or the lightest supersymmetric particle compose the dark matter in the universe. Non-thermal leptogenesis is naturally implemented in this class of models.

An application of approach theory to the relative Hausdorff measure of non-compactness for the Wasserstein metric

After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related to the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metric space has some nice stability properties. We then combine the obtained result with Prokhorov's Theorem to establish inequalities between the relative Hausdorff measure of non-compactness for the Wasserstein metric and a canonical measure of non-uniform integrability.

Exponential stability of matrix-valued Markov chains via nonignorable periodic data

Let ξ = { ξ n } n ≥ 0 \boldsymbol {\xi }=\{\xi _n\}_{n\ge 0} be a Markov chain defined on a probability space ( Ω , F , P ) (\Omega ,\mathscr {F},\mathbb {P}) valued in a discrete topological space S \boldsymbol {S} that consists of a finite number of real d × d d\times d matrices. As usual, ξ \boldsymbol {\xi } is called uniformly exponentially stable if there exist two constants C > 0 C>0 and 0 > λ > 1 0>\lambda >1 such that P ( ‖ ξ 0 ( ω ) ⋯ ξ n − 1 ( ω ) ‖ ≤ C λ n ∀ n ≥ 1 ) = 1 ; \begin{gather*} \mathbb {P}\left (\|\xi _0(\omega )\dotsm \xi _{n-1}(\omega )\|\le C\lambda ^{n}\ \forall n\ge 1\right )=1; \end{gather*} and ξ \boldsymbol {\xi } is called nonuniformly exponentially stable if there exist two random variables C ( ω ) > 0 C(\omega )>0 and 0 > λ ( ω ) > 1 0>\lambda (\omega )>1 such that P ( ‖ ξ 0 ( ω ) ⋯ ξ n − 1 ( ω ) ‖ ≤ C ( ω ) λ ( ω ) n ∀ n ≥ 1 ) = 1. \begin{gather*} \mathbb {P}\left (\|\xi _0(\omega )\dotsm \xi _{n-1}(\omega )\|\le C(\omega )\lambda (\omega )^{n}\ \forall n\ge 1\right )=1. \end{gather*} In this paper, we characterize the exponential stabilities of ξ \boldsymbol {\xi } via its nonignorable periodic data whenever ξ \boldsymbol {\xi } has a constant transition binary matrix. As an application, we construct a Lipschitz continuous S L ( 2 , R ) \mathrm {SL}(2,\mathbb {R}) -cocycle driven by a Markov chain with 2 2 -points state space, which is nonuniformly but not uniformly hyperbolic and which has constant Oseledeč splitting with respect to a canonical Markov measure.

The canonical measure on a reductive p-adic group is motivic

Let $G$ be a connected reductive group over a non-Archimedean local field. We prove that its parahoric subgroups are definable in the Denef-Pas language, which is a first-order language of logic used in the theory of motivic integration developed by Cluckers and Loeser. The main technical result is the definability of the connected component of the N\'eron model of a tamely ramified algebraic torus. As a corollary, we prove that the canonical Haar measure on $G$, which assigns volume $1$ to the particular \emph{canonical} maximal parahoric defined by Gross, is motivic. This result resolves a technical difficulty that arose in Cluckers-Gordon-Halupczok and Shin-Templier and permits a simplification of some of the proofs in those articles. It also allows us to show that formal degree of a compactly induced representation is a motivic function of the parameters defining the representation.

Equidistribution speed for Fekete points associated with an ample line bundle

Let K be the closure of a bounded open set with smooth boundary in C n . A Fekete configuration of order p for K is a finite subset of K maximizing the Vandermonde determinant associated with polynomials of degree p. A recent theorem by Berman, Boucksom and Witt Nystrimplies that Fekete configurations for K are asymptotically equidistributed with respect to a canonical equilibrium measure, as p!1. We give here an explicit estimate for the speed of convergence. The result also holds in a general setting of Fekete points associated with an ample line bundle over a projective manifold. Our approach requires a new estimate on Bergman kernels for line bundles and quantitative results in pluripotential theory which are of independent interest. Classification AMS 2010: 32U15 (32L05).

A criterion for convergence to super-Brownian motion on path space

We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the $r$-point functions for $r=2,\ldots,5$. The $r$-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the $r$-point functions, but the lack of tightness has been an obstruction to proving convergence on path space. We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.

Systematic Tail Risk

We propose new systematic tail risk measures constructed using two different approaches. The first extends the canonical downside beta and co-moment measures, while the second is based on the sensitivity of stock returns to innovations in market crash risk. Both tail risk measures are associated with a significantly positive risk premium after controlling for other measures of downside risk, including downside beta, co-skewness and co-kurtosis. Using these measures, we examine the relevance of the tail risk premium for investors with different investment horizons.

False Positives and Other Statistical Errors in Standard Analyses of Eye Movements in Reading.

In research on eye movements in reading, it is common to analyze a number of canonical dependent measures to study how the effects of a manipulation unfold over time. Although this gives rise to the well-known multiple comparisons problem, i.e. an inflated probability that the null hypothesis is incorrectly rejected (Type I error), it is accepted standard practice not to apply any correction procedures. Instead, there appears to be a widespread belief that corrections are not necessary because the increase in false positives is too small to matter. To our knowledge, no formal argument has ever been presented to justify this assumption. Here, we report a computational investigation of this issue using Monte Carlo simulations. Our results show that, contrary to conventional wisdom, false positives are increased to unacceptable levels when no corrections are applied. Our simulations also show that counter-measures like the Bonferroni correction keep false positives in check while reducing statistical power only moderately. Hence, there is little reason why such corrections should not be made a standard requirement. Further, we discuss three statistical illusions that can arise when statistical power is low, and we show how power can be improved to prevent these illusions. In sum, our work renders a detailed picture of the various types of statistical errors than can occur in studies of reading behavior and we provide concrete guidance about how these errors can be avoided.

An overlook on statistical inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition

ABSTRACTIn this paper, we are interested in nonparametric inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition. This condition relates the magnitude of the dissipative term and the magnitude of the random term. The precise balance between the drift term which removes energy in average and the stochastic term provided by the fluctuation-dissipation relation insures that the canonical measure is preserved by the dynamics. In this framework, it is possible to give an explicit construction of a Lyapunov function and thus to prove exponential ergodicity. Then, we consider various estimation procedures and provide also a numerical section, where simulations are conducted.

Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing

We unify and extend a number of approaches related to constructing multivariate Madan–Seneta Variance-Gamma models for option pricing. Complementing Grigelionis’ (2007) class, an overarching model is derived by subordinating multivariate Brownian motion to a subordinator from Thorin’s (1977) [58, 59] class of generalised Gamma convolutions. Multivariate classes developed by Pérez-Abreu and Stelzer (2014), Semeraro (2008) and Guillaume (2013) are submodels. The classes are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures are obtained, which permit applications such as option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.