scholarly journals On the Number of Solutions in Random Graphk-Colouring

2018 ◽  
Vol 28 (1) ◽  
pp. 130-158 ◽  
Author(s):  
FELICIA RASSMANN
Keyword(s):  
Phase Transition ◽  
Asymptotic Distribution ◽  
Random Graph ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Wide Range

Letk⩾ 3 be a fixed integer. We exactly determine the asymptotic distribution of lnZk(G(n, m)), whereZk(G(n, m)) is the number ofk-colourings of the random graphG(n, m). A crucial observation to this end is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles inG(n, m). Our result holds for a wide range of average degrees, and forkexceeding a certain constantk0it covers all average degrees up to the so-calledcondensation phase transition.

scholarly journals Planting Colourings Silently

2016 ◽  
Vol 26 (3) ◽  
pp. 338-366 ◽  
Author(s):  
VICTOR BAPST ◽  
AMIN COJA-OGHLAN ◽  
CHARILAOS EFTHYMIOU
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Random Variable ◽  
Average Degree ◽  
Fixed Integer ◽  
Wide Range ◽  
Concentration Result

Letk⩾ 3 be a fixed integer and letZk(G) be the number ofk-colourings of the graphG. For certain values of the average degree, the random variableZk(G(n,m)) is known to be concentrated in the sense that$\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability (Achlioptas and Coja-Oghlan,Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees,$\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$converges to 0 in probability foranydiverging function$\omega=\omega(n)\ra\infty$. Forkexceeding a certain constantk0this result covers all average degrees up to the so-calledcondensation phase transitiondk,con, and this is best possible. As an application, we show that the experiment of choosing ak-colouring of the random graphG(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

scholarly journals The Diameter of Sparse Random Graphs

2010 ◽  
Vol 19 (5-6) ◽  
pp. 835-926 ◽  
Author(s):  
OLIVER RIORDAN ◽  
NICHOLAS WORMALD
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Branching Process ◽  
Correction Term ◽  
Separate Analysis ◽  
Wide Range ◽  
Range Of Functions ◽  
Finite Distance ◽  
Additive Correction

In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ > 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

scholarly journals A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

2021 ◽  
Author(s):  
Sven-Erik Ekström ◽  
Paris Vassalos
Keyword(s):  
Distribution Function ◽  
Asymptotic Distribution ◽  
Numerical Experiments ◽  
Toeplitz Matrices ◽  
Future Research ◽  
Research Directions ◽  
Numerical Examples ◽  
Working Hypothesis ◽  
Wide Range ◽  
Future Research Directions

AbstractIt is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

Fabrication of nanocrystalline λ-Ti3O5with tunable terahertz wave transmission properties across a temperature induced phase transition

2016 ◽  
Vol 4 (43) ◽  
pp. 10279-10285 ◽  
Author(s):  
Qiwu Shi ◽  
Guoqing Chai ◽  
Wanxia Huang ◽  
Yanli Shi ◽  
Bo Huang ◽  
...  
Keyword(s):  
Phase Transition ◽  
Carbothermal Reduction ◽  
Terahertz Wave ◽  
Wave Transmission ◽  
Slow Phase ◽  
Dynamic Tuning ◽  
Transmission Properties ◽  
Wide Range

Nanocrystalline λ-Ti3O5was fabricated by carbothermal reduction of nano-TiO2. It exhibits a continuous and slow phase transition across a wide range of temperatures, which can lead to dynamic tuning of THz transmission.

scholarly journals Effect of Sintering Temperature on Metal-Insulator Phase Transition in La1-xCaxMnO3 Perovskites

2020 ◽  
Vol 2 (1) ◽  
pp. 37-42
Author(s):  
Arunachalam M ◽  
Thamilmaran P ◽  
Sakthipandi K
Keyword(s):  
Phase Transition ◽  
Bulk Modulus ◽  
Sintering Temperature ◽  
Magnetic Sensors ◽  
Temperature Dependent ◽  
Metal Insulator ◽  
Wide Range ◽  
Different Temperatures ◽  
Insulator Phase Transition

Lanthanum calcium based perovskites are found to be advantageous for the possible applications in magnetic sensors/reading heads, cathodes in solid oxide fuel cells, and frequency switching devices. In the present investigation La0.3Ca0.7MnO3 perovskites were synthesised through solid state reaction and sintered at four different temperatures such as 900, 1000, 1100 and 1200˚ C. X-ray powder diffraction pattern confirms that the prepared La0.3Ca0.7MnO3 perovskites have orthorhombic structure with Pnma space group. Ultrasonic in-situ measurements have been carried out on the La0.3Ca0.7MnO3 perovskites over wide range of temperature and elastic constants such as bulk modulus of the prepared La0.3Ca0.7MnO3 perovskites was obtained as function of temperature. The temperature-dependent bulk modulus has shown an interesting anomaly at the metal-insulator phase transition. The metal insulator transition temperature derived from temperature-dependent bulk modulus increases from temperature 352˚ C to 367˚ C with the increase of sintering temperature from 900 to 1200˚ C.

scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals Ударно-волновой полиморфный переход в пористом графите

2021 ◽  
Vol 91 (11) ◽  
pp. 1707
Author(s):  
С.А. Кинеловский
Keyword(s):  
Phase Transition ◽  
Shock Wave ◽  
Elastic Energy ◽  
Polymorphic Transformation ◽  
Polymorphic Transition ◽  
Initial Structure ◽  
Graphite Phase ◽  
Proposed Model ◽  
Porous Graphite ◽  
Wide Range

For the polymorphic transformation of porous graphite in the shock, the previously proposed model linking the process of graphite phase transition with a change in the elastic energy of a substance has been tested. It is shown that the model plausibly describes the experimental results outside the transition zone in a fairly wide range of changes in the porosity of samples with their different initial structure. It is discussed how the model under consideration changes the currently existing ideas about the thermodynamics of the polymorphic transition of matter in a shock wave.

Photocontrol of a dynamic phase transition in the CDW state

2002 ◽  
Vol 12 (9) ◽  
pp. 297-301
Author(s):  
K. Miyano ◽  
N. Ogawa
Keyword(s):  
Phase Transition ◽  
Internal Stress ◽  
One Dimensional ◽  
Dynamic Phase ◽  
Wide Range ◽  
Dynamic Phase Transition

A strong photoinduced suppression of the creep to slide dynamic phase transition was discovered in a quasi one-dimensional CDW material, K0.3MoO3. It is manifested in hysteretic “photoresistivity", an increase in the resistance by orders of magnitude under the illumination of light of modest intensity. Based on a wide range of measurements, it is concluded that the effect is due to the release of the internal stress by photoexcited jumps of the CDW phase over pinning sites: a phenomenon which can be effective only in a system, in which the thermally excited quasiparticles are negligibly small and hence the `bare' CDW properties emerge.

On General Polynomials

1967 ◽  
Vol 10 (4) ◽  
pp. 579-583 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Finite Field ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Image Position

Let d denote a fixed integer > 1 and let GF(q) denote the finite field of q = pn elements. We consider q fixed ≥ A(d), where A(d) is a (large) constant depending only on d. Let1where each aiεGF(q). Let nr(r = 2, 3, …, d) denote the number of solutions in GF(q) offor which x1, x2, …, xr are all different.

Sign in / Sign up

Export Citation Format

Share Document