Fluctuations of the free energy in the REM and the p-spin SK models

We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the $p$-spin version, we prove that (for $p$ even) the random corrections to the free energy are on a scale $N^{-(p-2)/4}$ only, and after proper rescaling converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, \b, smaller than a critical \b_p. We also show that \b_p\to \sqrt{2\ln 2} as $p\uparrow +\infty$. Additionally we study the formal $p\uparrow +\infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the partition function at {\it all} temperatures. For \b<\sqrt{2\ln2}, fluctuations are found at an {\it exponentially small} scale, with two distinct limit laws above and below a second critical value $\sqrt{\ln 2/2}$: For \b up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For \b larger than the critical $\sqrt{2\ln 2}$, the fluctuations of the logarithm of the partition function are on scale one and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1/2.Comment: 40pp, AMSTe

tt-Martin boundary of killed random walks in the quadrant

We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our approach is uniform in $t$, and shows that there are three regimes for the Martin boundary.Comment: 18 pages, 2 figures, to appear in S\'eminaire de Probabilit\'e

On the functions counting walks with small steps in the quarter plane

Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of ${\bf C}^2$, the interval $]0,1/|\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure

Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial'' compactification obtained by Ney and Spitzer for the homogeneous random walks in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times\N$ to a point of on the Martin boundary does not imply convergence of the sequence $z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large deviation properties of the scaled processes and uses Pascal's method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.Comment: 42 pages, preprint, CNRS UMR 808

ПАРАМЕТРЫ ЭКОЛОГИЧЕСКОЙ ПЛАСТИЧНОСТИ СОРТОВ И СОРТООБРАЗЦОВ ЯРОВОГО ЯЧМЕНЯ АМУРСКОЙ СЕЛЕКЦИИ

The article is concerned with increasing of crop yield and explores that production quality is influenced by the variety adjusted to local conditions. This variety is most productive for plant production and important in agricultural production. New cultivar should be highly productive, highly adaptive and environmentally plastic (to form steady crop yield in different conditions). The article explores estimation of cultivars and varieties of Amur spring barley on environmental plasticity and stability. The researchers estimated environmental plasticity and stability for 3 years (2012–2014), which differed in vegetation conditions. The authors apply regression co-efficient (bi), which characterize cultivars response to agricultural changes and stability variance (s2di), which shows cultivar response to environmental changes and its stability. New Amur variety included into the State List of Selection Inventions is not stable, which is proved by estimation in 2008–2011. Earlier it was not stable but responded well to the changes; now it is not stable but more productive in favorable conditions. The authors make the idea that varieties, which belong to the group of well-responding to the changes and stable ones are the most significant varieties. The researchers define Mishka variety as a stable and well-responding. Огромную роль в повышении урожайности и улучшении качества продукции играет сорт, приспособленный к местным условиям. Он является основой производства любой растениеводческой продукции и его роль в сельскохозяйственном производстве постоянно возрастает. Новый сорт должен быть не только высокоурожайным, но обладать высокой адаптивной способностью и широкой экологической пластичностью (формировать стабильный урожай в различных условиях). Статья посвящена вопросу оценки сортов и сортообразцов ярового ячменя амурской селекции по параметрам экологической пластичности и стабильности. Расчет экологической пластичности и стабильности проводили в среднем за 3 года (2012–2014 гг.), сильно отличающиеся по условиям вегетации. Для определения данных параметров приведен расчет коэффициента регрессии (bi ), характеризующего реакцию сортов на изменение условий выращивания, и вариансы стабильности (s 2 di), которая указывает, насколько сорт отзывчив на условия среды и стабилен ли в этих условиях. Новый сорт амурской селекции Амур, внесенный в Государственный реестр селекционных достижений в 2015 г., является нестабильным, что также подтверждается и ранее проведенными расчетами (в 2008–2011 гг.). Если он ранее был нестабильным, но хорошо отзывчивым на изменение условий, то в данный момент он характеризуется как нестабильный и показывающий лучшие результаты в благоприятных условиях. Наибольшее значение имеют сорта, которые относятся к группе хорошо отзывчивых на изменение условий и являются стабильными. Из изученных нами 12 сортообразцов к этой группе можно отнести один – сортообразец Мишка

Analysis of the Karmarkar-Karp Differencing Algorithm

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling $n^{-c\ln n}$, where $c=1/(2\ln2)=0.7213...$. Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.Comment: 9 pages, 8 figures; minor change

Mass measurements of As, Se and Br nuclei and their implication on the proton-neutron interaction strength towards the N=Z line

Mass measurements of the nuclides 69As, 70,71Se, and 71Br, produced via fragmentation of a 124Xe primary beam at the Fragment Separator (FRS) at GSI, have been performed with the multiple-reflection time-of-flight mass spectrometer (MR-TOF-MS) of the FRS Ion Catcher with an unprecedented mass resolving power of almost 1000000. Such high resolving power is the only way to achieve accurate results and resolve overlapping peaks of short-lived exotic nuclei, whose total number of accumulated events is always limited. For the nuclide 69As, this is the first direct mass measurement. A mass uncertainty of 22 keV was achieved with only ten events. For the nuclide 70Se, a mass uncertainty of 2.6 keV was obtained, corresponding to a relative accuracy of δm/m=4.0×10−8, with less than 500 events. The masses of the nuclides 71Se and 71Br have been measured with an uncertainty of 23 and 16 keV, respectively. Our results for the nuclides 70,71Se and 71Br are in good agreement with the 2016 Atomic Mass Evaluation, and our result for the nuclide 69As resolves the discrepancy between the previous indirect measurements. We measured also the mass of the molecule 14N15N40Ar (A=69) with a relative accuracy of δm/m=1.7×10−8, the highest yet achieved with an MR-TOF-MS. Our results show that the measured restrengthening of the proton-neutron interaction (δVpn) for odd-odd nuclei along the N=Z line above Z=29 (recently extended to Z=37) is hardly evident at the N−Z=2 line, and not evident at the N−Z=4 line. Nevertheless, detailed structure of δVpn along the N−Z=2 and N−Z=4 lines, confirmed by our mass measurements, may provide a hint regarding the ongoing ≈500 keV discrepancy in the mass value of the nuclide 70Br, which prevents including it in the world average of Ft value for superallowed 0+→0+β decays. The reported work sets the stage for mass measurements with the FRS Ion Catcher of nuclei at and beyond the N=Z line in the same region of the nuclear chart, including the nuclide 70Br.peerReviewe

ATHENA: A knowledge-based hybrid backpropagation-grammatical evolution neural network algorithm for discovering epistasis among quantitative trait Loci

<p>Abstract</p> <p>Background</p> <p>Growing interest and burgeoning technology for discovering genetic mechanisms that influence disease processes have ushered in a flood of genetic association studies over the last decade, yet little heritability in highly studied complex traits has been explained by genetic variation. Non-additive gene-gene interactions, which are not often explored, are thought to be one source of this "missing" heritability.</p> <p>Methods</p> <p>Stochastic methods employing evolutionary algorithms have demonstrated promise in being able to detect and model gene-gene and gene-environment interactions that influence human traits. Here we demonstrate modifications to a neural network algorithm in ATHENA (the Analysis Tool for Heritable and Environmental Network Associations) resulting in clear performance improvements for discovering gene-gene interactions that influence human traits. We employed an alternative tree-based crossover, backpropagation for locally fitting neural network weights, and incorporation of domain knowledge obtainable from publicly accessible biological databases for initializing the search for gene-gene interactions. We tested these modifications <it>in silico </it>using simulated datasets.</p> <p>Results</p> <p>We show that the alternative tree-based crossover modification resulted in a modest increase in the sensitivity of the ATHENA algorithm for discovering gene-gene interactions. The performance increase was highly statistically significant when backpropagation was used to locally fit NN weights. We also demonstrate that using domain knowledge to initialize the search for gene-gene interactions results in a large performance increase, especially when the search space is larger than the search coverage.</p> <p>Conclusions</p> <p>We show that a hybrid optimization procedure, alternative crossover strategies, and incorporation of domain knowledge from publicly available biological databases can result in marked increases in sensitivity and performance of the ATHENA algorithm for detecting and modelling gene-gene interactions that influence a complex human trait.</p