Spectrally degenerate graphs: Hereditary case

It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally, for all d-degenerate graphs, where the corresponding upper bound is sqrt{4dD}. Following this, we say that a graph G is spectrally d-degenerate if every subgraph H of G has spectral radius at most sqrt{d.Delta(H)}. In this paper we derive a rough converse of the above-mentioned results by proving that each spectrally d-degenerate graph G contains a vertex whose degree is at most 4dlog_2(D/d) (if D>=2d). It is shown that the dependence on D in this upper bound cannot be eliminated, as long as the dependence on d is subexponential. It is also proved that the problem of deciding if a graph is spectrally d-degenerate is co-NP-complete.Comment: Updated after reviewer comments. 14 pages, no figure

Ultrafast carrier relaxation in GaN, In_(0.05)Ga_(0.95)N and an In_(0.05)Ga_(0.95)/In_(0.15)Ga_(0.85)N Multiple Quantum Well

Room temperature, wavelength non-degenerate ultrafast pump/probe measurements were performed on GaN and InGaN epilayers and an InGaN multiple quantum well structure. Carrier relaxation dynamics were investigated as a function of excitation wavelength and intensity. Spectrally-resolved sub-picosecond relaxation due to carrier redistribution and QW capture was found to depend sensitively on the wavelength of pump excitation. Moreover, for pump intensities above a threshold of 100 microJ/cm2, all samples demonstrated an additional emission feature arising from stimulated emission (SE). SE is evidenced as accelerated relaxation (< 10 ps) in the pump-probe data, fundamentally altering the re-distribution of carriers. Once SE and carrier redistribution is completed, a slower relaxation of up to 1 ns for GaN and InGaN epilayers, and 660 ps for the MQW sample, indicates carrier recombination through spontaneous emission.Comment: submitted to Phys. Rev.

Exotic local limit theorems at the phase transition in free products

We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that $\mu$ * n (e) $\sim$ CR --n n --5/3 if d = 5 and $\mu$ * n (e) $\sim$ CR --n n --3/2 log(n) --1/2 if d = 6, where $\mu$ * n is the nth convolution power of $\mu$ and R is the inverse of the spectral radius of $\mu$. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a rst version of [11]. This also shows that the classication of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete

Spontaneous Pattern Formation in a Polariton Condensate

Polariton condensation can be regarded as a self-organization phenomenon, where phase ordering is established among particles in the system. In such condensed systems, further ordering can possibly occur in the particle density distribution, under particular experimental conditions. In this work we report on spontaneous pattern formation in a polariton condensate under non-resonant optical pumping. The slightly elliptical ring-shaped excitation laser we employ is such to force condensation to occur in a single-energy state with periodic boundary conditions, giving rise to a multi-lobe standing wave patterned state