On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

Soft bounds on diffusion produce skewed distributions and Gompertz growth

Constraints can affect dramatically the behavior of diffusion processes. Recently, we analyzed a natural and a technological system and reported that they perform diffusion-like discrete steps displaying a peculiar constraint, whereby the increments of the diffusing variable are subject to configuration-dependent bounds. This work explores theoretically some of the revealing landmarks of such phenomenology, termed "soft bound". At long times, the system reaches a steady state irreversibly (i.e., violating detailed balance), characterized by a skewed "shoulder" in the density distribution, and by a net local probability flux, which has entropic origin. The largest point in the support of the distribution follows a saturating dynamics, expressed by the Gompertz law, in line with empirical observations. Finally, we propose a generic allometric scaling for the origin of soft bounds. These findings shed light on the impact on a system of such "scaling" constraint and on its possible generating mechanisms.Comment: 9 pages, 6 color figure

Characterizing Pixel and Point Patterns with a Hyperuniformity Disorder Length

We introduce the concept of a hyperuniformity disorder length that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size. In particular, fluctuations are determined by the average number of particles within a distance $h$ from the boundary of the window. We first compute special expectations and bounds in $d$ dimensions, and then illustrate the range of behavior of $h$ versus window size $L$ by analyzing three different types of simulated two-dimensional pixel pattern - where particle positions are stored as a binary digital image in which pixels have value zero/one if empty/contain a particle. The first are random binomial patterns, where pixels are randomly flipped from zero to one with probability equal to area fraction. These have long-ranged density fluctuations, and simulations confirm the exact result $h=L/2$. Next we consider vacancy patterns, where a fraction $f$ of particles on a lattice are randomly removed. These also display long-range density fluctuations, but with $h=(L/2)(f/d)$ for small $f$. For a hyperuniform system with no long-range density fluctuations, we consider Einstein patterns where each particle is independently displaced from a lattice site by a Gaussian-distributed amount. For these, at large $L$, $h$ approaches a constant equal to about half the root-mean-square displacement in each dimension. Then we turn to grayscale pixel patterns that represent simulated arrangements of polydisperse particles, where the volume of a particle is encoded in the value of its central pixel. And we discuss the continuum limit of point patterns, where pixel size vanishes. In general, we thus propose to quantify particle configurations not just by the scaling of the density fluctuation spectrum but rather by the real-space spectrum of $h(L)$ versus $L$. We call this approach Hyperuniformity Disorder Length Spectroscopy

Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees

We develop a technique that provides a lower bound on the speed of transient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and regeneration time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic (2002) prove an upper bound of the form $b/(b+\delta)$ for the speed on the $b$-ary tree, where $\delta$ is the reinforcement parameter. For $\delta>1$ we provide a lower bound of the form $\gamma^2 b/(b+\delta)$, where $\gamma$ is the survival probability of an associated branching process.Comment: 21 page

Multiple-Access Bosonic Communications

The maximum rates for reliably transmitting classical information over Bosonic multiple-access channels (MACs) are derived when the transmitters are restricted to coherent-state encodings. Inner and outer bounds for the ultimate capacity region of the Bosonic MAC are also presented. It is shown that the sum-rate upper bound is achievable with a coherent-state encoding and that the entire region is asymptotically achievable in the limit of large mean input photon numbers.Comment: 11 pages, 5 figures, corrected two figures, accepted for publication in Phys. Rev.