Accessibility percolation with backsteps

Consider a graph in which each site is endowed with a value called \emph{fitness}. A path in the graph is said to be "open" or "accessible" if the fitness values along that path is strictly increasing. We say that there is accessibility percolation between two sites when such a path between them exists. Motivated by the so called House-of-Cards model from evolutionary biology, we consider this question on the $L$-hypercube $\{0,1\}^L$ where the fitness values are independent random variables. We show that, in the large $L$ limit, the probability that an accessible path exists from an arbitrary starting point to the (random) fittest site is no more than $x^*_{1/2}= 1-\frac12\sinh^{-1}(2) =0.27818\ldots$ and we conjecture that this probability does converge to $x^*_{1/2}$. More precisely, there is a phase transition on the value of the fitness $x$ of the starting site: assuming that the fitnesses are uniform in $[0,1]$, we show that, in the large $L$ limit, there is almost surely no path to the fittest site if $x>x^*_{1/2}$ and we conjecture that there are almost surely many paths if $x<x^*_{1/2}$. If one conditions on the fittest site to be on the opposite corner of the starting site rather than being randomly chosen, the picture remains the same but with the critical point being now $x^*_1= 1-\sinh^{-1}(1)= 0.11863\ldots$. Along the way, we obtain a large $L$ estimation for the number of self-avoiding paths joining two opposite corners of the $L$-hypercube

The number of accessible paths in the hypercube

Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube $\{0,1\}^L$ where each node carries an independent random variable uniformly distributed on $[0,1]$, except $(1,1,\ldots,1)$ which carries the value $1$ and $(0,0,\ldots,0)$ which carries the value $x\in[0,1]$. We study the number $\Theta$ of paths from vertex $(0,0,\ldots,0)$ to the opposite vertex $(1,1,\ldots,1)$ along which the values on the nodes form an increasing sequence. We show that if the value on $(0,0,\ldots,0)$ is set to $x=X/L$ then $\Theta/L$ converges in law as $L\to\infty$ to $\mathrm{e}^{-X}$ times the product of two standard independent exponential variables. As a first step in the analysis, we study the same question when the graph is that of a tree where the root has arity $L$, each node at level 1 has arity $L-1$, \ldots, and the nodes at level $L-1$ have only one offspring which are the leaves of the tree (all the leaves are assigned the value 1, the root the value $x\in[0,1]$).Comment: Published at http://dx.doi.org/10.3150/14-BEJ641 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

We study a dyadic branching Brownian motion on the real line with absorption at 0, drift $\mu \in \mathbb{R}$ and started from a single particle at position $x>0.$ When $\mu$ is large enough so that the process has a positive probability of survival, we consider $K(t),$ the number of individuals absorbed at 0 by time $t$ and for $s\ge 0$ the functions $\omega_s(x):= \mathbb{E}^x[s^{K(\infty)}].$ We show that $\omega_s<\infty$ if and only of $s\in[0,s_0]$ for some $s_0>1$ and we study the properties of these functions. Furthermore, for $s=0, \omega(x) := \omega_0(x) =\mathbb{P}^x(K(\infty)=0)$ is the cumulative distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption. We give three descriptions of the family $\omega_s, s\in [0,s_0]$ through a single pair of functions, as the two extremal solutions of the Kolmogorov-Petrovskii-Piskunov (KPP) traveling wave equation on the half-line, through a martingale representation and as an explicit series expansion. We also obtain a precise result concerning the tail behavior of $K(\infty)$. In addition, in the regime where $K(\infty)>0$ almost surely, we show that $u(x,t) := \mathbb{P}^x(K(t)=0)$ suitably centered converges to the KPP critical travelling wave on the whole real line.Comment: Grant information adde

Policies and Deployment for Fuel Cell Electric Vehicles An Assessment of the Normandy Project

The paper provides a cost benefit analysis of one of the most prominent deployment project in France of fuel cell electric vehicles, taking place in Normandy. The project builds on the substitution of a diesel Renault Kangoo by an electric Renault Kangoo ZE with afuel cell range extender for public fleets. The analysis points out potential weaknesses of the project as it is envisioned today using a decomposition of the value-chain. To achieve sustainability in 2025 a much stronger deployment should take place. This would allow for a sharp decrease in the total cost of ownership thanks to a close coordination between hydrogen production and its delivery through refilling stations to take advantage of the expected increasing volume of hydrogen consumption along the deployment path. Thissuggests that a high level in public funds at this early deployment phase can be critical for the success of the project

Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.Comment: 56 pages, 1 figur

Light-controlled flows in active fluids

International audienceMany photosynthetic microorganisms are able to detect light and move toward optimal intensities. This ability, known as phototaxis, plays a major role in ecology by affecting natural phytoplankton mass transfers and has important applications in bioreactor and artificial microswimmers technologies. Here we show that this property can be exploited to generate macroscopic fluid flows using a localized light source directed toward shallow suspensions of phototactic microorganisms. Within the intensity range of positive phototaxis, algae accumulate beneath the excitation light where collective effects lead to the emergence of radially symmetric convective flows. These flows can thus be used as hydrodynamic tweezers to manipulate small floating objects. At high cell density and layer depth, we uncover a new kind of instability wherein the viscous torque exerted by self-generated fluid flows on the swimmers induces the formation of traveling waves. A model coupling fluid flow, cell concentration and orientation finely reproduces the experimental data

Global existence for a free boundary problem of Fisher-KPP type

Motivated by the study of branching particle systems with selection, we establish global existence for the solution $(u,\mu)$ of the free boundary problem $$\begin{cases} \partial_t u =\partial^2_{x} u +u & \text{for $t>0$ and $x>\mu_t$,}\\ u(x,t)=1 &\text{for $t>0$ and $x \leq \mu_t$}, \\ \partial_x u(\mu_t,t)=0 & \text{for $t>0$}, \\ u(x,0)=v(x) &\text{for $x\in \mathbb{R}$}, \end{cases}$$ when the initial condition $v:\mathbb{R}\to[0,1]$ is non-increasing with $v(x) \to 0$ as $x\to \infty$ and $v(x)\to 1$ as $x\to -\infty$. We construct the solution as the limit of a sequence $(u_n)_{n\ge 1}$, where each $u_n$ is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi \textit{et al.}~\cite{DeMasi2017a} show that this global solution can be identified with the hydrodynamic limit of the so-called $N$-BBM, {\it i.e.} a branching Brownian motion in which the population size is kept constant equal to $N$ by killing the leftmost particle at each branching event

Irreversible Collective Migration of Cyanobacteria in Eutrophic Conditions

In response to natural or anthropocentric pollutions coupled to global climate changes, microorganisms from aquatic environments can suddenly accumulate on water surface. These dense suspensions, known as blooms, are harmful to ecosystems and significantly degrade the quality of water resources. In order to determine the physico-chemical parameters involved in their formation and quantitatively predict their appearance, we successfully reproduced irreversible cyanobacterial blooms in vitro. By combining chemical, biochemical and hydrodynamic evidences, we identify a mechanism, unrelated to the presence of internal gas vesicles, allowing the sudden collective upward migration in test tubes of several cyanobacterial strains (Microcystis aeruginosa PCC 7005, Microcystis aeruginosa PCC 7806 and Synechocystis sp. PCC 6803). The final state consists in a foamy layer of biomass at the air-liquid interface, in which micro-organisms remain alive for weeks, the medium lying below being almost completely depleted of cyanobacteria. These "laboratory blooms" start with the aggregation of cells at high ionic force in cyanobacterial strains that produce anionic extracellular polymeric substances (EPS). Under appropriate conditions of nutrients and light intensity, the high photosynthetic activity within cell clusters leads the dissolved oxygen (DO) to supersaturate and to nucleate into bubbles. Trapped within the EPS, these bubbles grow until their buoyancy pulls the biomass towards the free surface. By investigating a wide range of spatially homogeneous environmental conditions (illumination, salinity, cell and nutrient concentration) we identify species-dependent thresholds and timescales for bloom formation. We conclude on the relevance of such results for cyanobacterial bloom formation in the environment and we propose an efficient method for biomass harvesting in bioreactors.Comment: 16 Pages, 4 figure

The deployment of BEV and FCEV in 2015

In Europe the transport sector contributes about 25% of total GHG emissions, 75% of which come from road transport. Contrarily to industrial emissions road emissions have increased over the period 1990-2015 in OECD countries: California (+26%), Germany (0%), France (+12%), Japan (+2%), Denmark (+30%). The number of registered vehicles on road in these countries amounts respectively to: California (33 million), Germany (61.5 million), France (38 million), Japan (77 million), Denmark (4 million). Even if these numbers are not expected to grow in the future this calls for major programs to reduce the corresponding GHG emissions in order to achieve the global GHG targets for 2050. The benefits from these programs will spread out to non OECD countries in which road emissions are bound to increase. Programs to promote zero emissions vehicles (ZEV) effectively started in the 2000’s through public private partnerships involving government agencies, manufacturers, utilities and fuel companies. These partnerships provided subsidies for R&D, pilot programs and infrastructure. Moreover, technical norms for emissions, global requirements for the portfolio of sales for manufacturers, rebates on the purchasing price for customers as well as various perks (driving bus lanes, free parking, etc.) are now in place. These multiple policy instruments constitute powerful incentives to orient the strategies of manufacturers and to stimulate the demand for ZEV. The carbon tax on the distribution of fossil fuels, whenever it exists, remains low and, at this stage, cannot be considered as an important driving force. The cases studies reveal important differences for the deployment of battery electric vehicle (BEV) versus fuel cell electric vehicle (FCEV). BEV is leading the game with a cheaper infrastructure investment cost and a lower cost for vehicle. The relatively low autonomy makes BEV mostly suited for urban use, which is a large segment of the road market. The current level of BEV vehicles on roads starts to be significant with California (70,000), Germany (25,000), France (31,000), Japan (608,000) Denmark (3,000), but they remain very low relative to the targets for 2020: California (1.5 million), Germany (1 million), France (2 million), Japan (0.8-1.1 million for ZEV new registrations), Denmark (0.25 million). The developments and efficiency gains in battery technology along with subsidies for battery charging public stations are expected to facilitate the achievement of the growth. The relative rates of equipment (number of publicly available stations / number of BEV) provide indirect evidence on the effort made in the different countries: California (3%), Germany (12%), France (28%), Japan (11%), and Denmark (61%). In some countries public procurement plays a significant role. In France Autolib (publicly available cars in towns) represents a large share of the overall BEV deployment (12%), and the government recently announced a 50% target for low emissions in all public vehicles new equipment. FCEV is still in an early deployment stage due to a higher infrastructure investment cost and a higher cost for vehicle. The relatively high autonomy combined with speed refueling make FCEV mostly suited for long distance and interurban usage. At present there are only a very limited numbers of HRS deployed: California (28), Germany (15), France (6), Japan (31), Japan (7), Denmark (7), and only a few units of H2 vehicles on road: California (300), Germany (125), France (60), Japan (7), Denmark (21). However, a detailed analysis of the current road maps suggests that FCEV has a large potential. Targets for the 2025-2030 horizons are significant in particular in Germany (4% in 2030), Denmark (4.5% in 2025) and Japan (15-20% for ZEV new registrations in 2020). The California ARB has recently redefined its program (subsidies and mandates) to provide higher incentives for FCEV. France appears to focus on specialized regional submarkets to promote FCEV (such as the use of H2 range extending light utility vehicles). The financing of the H2 infrastructure appears as a bottleneck for FCEV deployment. Roadmaps address this issue through progressive geographical expansion (clusters) and a high level of public subsidies hydrogen refueling station (HRS) in particular in all countries except France. At this stage of BEV and FCEV do not appear as direct competitors; they address distinct market segments. Unexpected delays in the development of infrastructure in FCEV, possible breakthroughs in battery technology, and the promotion of national champions may change the nature of this competition, making it more intense in the future

Brownian bees in the infinite swarm limit

The Brownian bees model is a branching particle system with spatial selection. It is a system of N particles which move as independent Brownian motions in Rd and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that, as N→∞, the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper (Trans. Amer. Math. Soc. 374 (2021) 6269–6329)), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds; that is, that as N→∞, the equilibrium density of the particle system converges to the steady-state solution of the free boundary problem