Linearly bounded infinite graphs

Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs

Anomalous dimension of transverse momentum broadening in planar N=4\mathcal{N}=4 SYM

The typical transverse momentum $Q_s(t)$ (or "saturation" momentum) acquired by a hard particle propagating through a $\mathcal{N}=4$ SYM plasma increases over time like $t^\gamma$, with an anomalous exponent $\gamma>1/2$ characteristic of super-diffusion. This anomalous exponent is a function of the 't Hooft coupling $\lambda=g^2N_c$. Recently, a method has been proposed to systematically compute the perturbative series of $\gamma(\lambda)$ at weak coupling. This method relies on the traveling wave interpretation of the time evolution of $Q_s(t)$ and on the dominance of soft-collinear radiative corrections at large times. In this paper, we compute $\gamma(\lambda)$ up to $\mathcal{O}(\lambda^{2})$ using the double logarithmic behaviour of the BFKL equation in planar $\mathcal{N}=4$ SYM at three loops. This calculation allows us to discuss the transition towards the strong coupling regime where AdS/CFT calculations predict $\gamma\to 1$.Comment: 8 pages, 1 figure, contribution to the proceedings of the "XVth International Conference on Quark Confinement and the Hadron Spectrum

Towards a cosmological neutrino mass detection

Future cosmological measurements should enable the sum of neutrino masses to be determined indirectly through their effects on the expansion rate of the Universe and the clustering of matter. We consider prospects for the gravitationally lensed Cosmic Microwave Background anisotropies and Baryon Acoustic Oscillations in the galaxy distribution, examining how the projected uncertainty of $\approx15$ meV on the neutrino mass sum (a 4$\sigma$ detection of the minimal mass) might be reached over the next decade. The current 1$\sigma$ uncertainty of $\approx 103$ meV (Planck-2015+BAO-15) will be improved by upcoming 'Stage-3' CMB experiments (S3+BAO-15: 44 meV), then upcoming BAO measurements (S3+DESI: 22 meV), and planned next-generation 'Stage 4' CMB experiments (S4+DESI: 15-19 meV, depending on angular range). An improved optical depth measurement is important: the projected neutrino mass uncertainty increases to $26$ meV if S4 is limited to $\ell>20$ and combined with current large-scale polarization data. Looking beyond $\Lambda$CDM, including curvature uncertainty increases the forecast mass error by $\approx$ 50% for S4+DESI, and more than doubles the error with a two-parameter dark energy equation of state. Complementary low-redshift probes including galaxy lensing will play a role in distinguishing between massive neutrinos and a departure from a $w=-1$, flat geometry.Comment: Submitted to PRD. 15 pages, 10 figure

Universality aspects of quantum corrections to transverse momentum broadening in QCD media

We study non-linear quantum corrections to transverse momentum broadening (TMB) of a fast parton propagating in dense QCD matter in the leading logarithmic approximation. These non-local corrections yield an anomalous super-diffusive behavior characterized by a heavy tailed distribution which is associated with L\'{e}vy random walks. Using a formal analogy with the physics of traveling waves, we show that at late times the transverse momentum distribution tends to a universal scaling regime. We derive analytic solutions in terms of an asymptotic expansion around the scaling limit for both fixed and running coupling. We note that our analytic approach yields a good agreement with the exact numerical solutions down to realistic values of medium length. Finally, we discuss the interplay between system size and energy dependence of the diffusion coefficient $\hat q$ and its connection with the gluon distribution function that is manifest at large transverse momentum transfer.Comment: 57 pages, 15 figure

Transverse momentum broadening from NLL BFKL to all orders in pQCD

We study, to all orders in perturbative QCD, the universal behavior of the saturation momentum $Q_s(L)$ controlling the transverse momentum distribution of a fast parton propagating through a dense QCD medium with large size $L$. Due to the double logarithmic nature of the quantum evolution of the saturation momentum, its large $L$ asymptotics is obtained by slightly departing from the double logarithmic limit of either next-to-leading log (NLL) BFKL or leading order DGLAP evolution equations. At fixed coupling, or in conformal $\mathcal{N}=4$ SYM theory, we derive the large $L$ expansion of $Q_s(L)$ up to order $\alpha_s^{3/2}$. In QCD with massless quarks, where conformal symmetry is broken by the running of the strong coupling constant, the one-loop QCD $\beta$-function fully accounts for the universal terms in the $Q_s(L)$ expansion. Therefore, the universal coefficients of this series are known exactly to all orders in $\alpha_s$.Comment: 13 pages, 4 figures, 3 appendice

Anomalous diffusion in QCD matter

We study the effects of quantum corrections on transverse momentum broadening of a fast parton passing through dense QCD matter. We show that, at leading logarithmic accuracy the broadening distribution tends at late times or equivalently for large system sizes $L$ to a universal distribution that only depends on a single scaling variable $k^2_\perp/Q^2_s$ where the typical transverse momentum scale increases with time as $\ln Q_s^2 \simeq (1+2 \beta ) \ln L - \frac{3}{2}(1+\beta )\,\ln\ln L$ up to non-universal terms, with an anomalous dimension $\beta \sim \sqrt{\alpha_s}$. This property is analogous to geometric scaling of gluon distributions in the saturation regime and traveling waves solutions to reaction-diffusion processes. We note that since $\beta >0$ the process is super-diffusive, which is also reflected at large transverse momentum where the scaling distribution exhibits a heavy tail $k_\perp^{4-2\beta }$ akin to L\'{e}vy random walks.Comment: 10 pages, 3 figures, 3 supplemental material

Anomalous dimension of transverse momentum broadening in planar = 4 SYM

The typical transverse momentum Qs(t) (or "saturation" momentum) acquired by a hard particle propagating through a N = 4 SYM plasma increases over time like tγ, with an anomalous exponent γ > 1/2 characteristic of super-diffusion. This anomalous exponent is a function of the ’t Hooft coupling λ = g2Nc. Recently, a method has been proposed to systematically compute the perturbative series of γ(λ) at weak coupling. This method relies on the traveling wave interpretation of the time evolution of Qs(t) and on the dominance of softcollinear radiative corrections at large times. In this paper, we compute γ(λ) up to (λ2) using the double logarithmic behaviour of the BFKL equation in planar = 4 SYM at three loops. This calculation allows us to discuss the transition towards the strong coupling regime where AdS/CFT calculations predict γ→1

On the regular structure of prefix rewritings

Projet MICASWe can consider a pushdown automaton as a word rewriting system with labelled rules applied only in a prefix way. The notion of context-free graph, defined by Muller and Schupp is then extended to the notion of prefix transition graph of a word rewriting system. Prefix transition graphs are context-free graphs, and we show they are also the rooted pattern graphs of finite degree, where a pattern graph produced from a finite graph by iterating the addition of a finite family of finite graphs (the patterns). Furthermore, this characterisation is effective in the following sense : any finite family of patterns generating a graph G having a finite degree and a root, is mapped effectively into a rewriting system R on words such that the prefix transition graph of R is isomorphic to G, and the reverse transformation is effective

Базовый алгоритм действия системы поддержки принятия решений

We consider two-player parity games played on transition graphs of higher order pushdown automata. They are ``game-equivalent'' to a kind of model-checking game played on graphs of the infinite hierarchy introduced recently by Caucal. Then in this hierarchy we show how to reduce a game to a graph of lower level. This leads to an effective solution and a construction of the winning strategies