Convergence to Equilibrium of Logit Dynamics for Strategic Games

We present the first general bounds on the mixing time of the Markov chain associated to the logit dynamics for wide classes of strategic games. The logit dynamics with inverse noise beta describes the behavior of a complex system whose individual components act selfishly and keep responding according to some partial ("noisy") knowledge of the system, where the capacity of the agent to know the system and compute her best move is measured by the inverse of the parameter beta. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that, for potential games, the mixing time is upper and lower bounded by an exponential in the inverse of the noise and in the maximum potential difference. Instead, for games with dominant strategies, the mixing time cannot grow arbitrarily with the inverse of the noise. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, a class of games that have been previously studied in Physics and, more recently, in Computer Science in the context of diffusion of new technologies. We give evidence that the mixing time of the logit dynamics for these games strongly depends on the structure of the underlying graph. We prove that the mixing time of the logit dynamics for these games can be upper bounded by a function that is exponential in the cutwidth of the underlying graph and in the inverse of noise. Moreover, we consider two specific and popular network topologies, the clique and the ring. For games played on a clique we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in the inverse of the noise and in the maximum potential difference, while for games played on a ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter

Particle-physics constraints on multifractal spacetimes

We study electroweak interactions in the multiscale theory with $q$-derivatives, a framework where spacetime has the typical features of a multifractal. In the simplest case with only one characteristic time, length and energy scale $t_*$, $\ell_*$, and $E_*$, we consider (i) the muon decay rate and (ii) the Lamb shift in the hydrogen atom, and constrain the corrections to the ordinary results. We obtain the independent absolute upper bounds (i) $t_* 35\,\text{MeV}$. Under some mild theoretical assumptions, the Lamb shift alone yields the even tighter ranges $t_*<10^{-27}\,{\rm s}$, $\ell_*<10^{-19}\,{\rm m}$, and $E_*>450\,\text{GeV}$. To date, these are the first robust constraints on the scales at which the multifractal features of the geometry can become important in a physical process.Comment: 5 pages. v2: units of derived bounds corrected, direct bounds and conclusions unchanged; v3: minor typos corrected to match the published versio

Poincar\'e profiles of groups and spaces

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincar\'{e} profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these profiles and conformal dimension. As applications, we use these invariants to show the non-existence of coarse embeddings in a variety of examples.Comment: 55 pages. To appear in Revista Matem\'atica Iberoamerican

New Standard Model constraints on the scales and dimension of spacetime

Using known estimates for the kaon--antikaon transitions, the mean lifetime of the muon and the mean lifetime of the tau, we place new and stronger constraints on the scales of the multi-fractional theories with weighted and $q$-derivatives. These scenarios reproduce a quantum-gravity regime where fields live on a continuous spacetime with a scale-dependent Hausdorff dimension. In the case with weighted derivatives, constraints from the muon lifetime are various orders of magnitude stronger than those from the tau lifetime and the kaon--antikaon transitions. The characteristic energy scale of the theory cannot be greater than $E_*>3\times 10^2\,{\rm TeV}$, and is tightened to $E_*>9\times 10^{8}\,{\rm TeV}$ for the typical value $\alpha=1/2$ of the fractional exponents in the spacetime measure. We also find an upper bound $d_{\rm H}<2.9$ on the spacetime Hausdorff dimension in the ultraviolet. In the case with $q$-derivatives, the strongest bound comes from the tau lifetime, but it is about 10 orders of magnitude weaker than for the theory with weighted derivatives.Comment: 1+21 pages, 3 figures, 1 table. v2: reference adde

Multi-wavelength signals of dark matter annihilations at the Galactic center

We perform a systematic study of the multi-wavelength signal induced by weakly interacting massive particle (WIMP) annihilations at the Galactic Center (GC). Referring to a generic WIMP dark matter (DM) scenario and depending on astrophysical inputs, we discuss spectral and angular features and sketch correlations among signals in the different energy bands. None of the components which have been associated to the GC source Sgr A*, nor the diffuse emission components from the GC region, have spectral or angular features typical of a DM source. Still, data-sets at all energy bands, namely, the radio, near infrared, X-ray and gamma-ray bands, contribute to place significant constraints on the WIMP parameter space. In general, the gamma-ray energy range is not the one with the largest signal to background ratio. In the case of large magnetic fields close to the GC, X-ray data give the tightest bounds. The emission in the radio-band, which is less model dependent, is very constraining as well. The recent detection by HESS of a GC gamma-ray source, and of a diffuse gamma-ray component, limits the possibility of a DM discovery with next generation of gamma-ray telescopes, like GLAST and CTA. We find that the most of the region in the parameter space accessible to these instruments is actually already excluded at other wave-lenghts. On the other hand, there may be still an open window to improve constraints with wide-field radio observations.Comment: 26 pages, 32 figures, treatments of starlight and interstellar medium improved, other minor changes, references adde