# Workshop Programme

## for period 26 - 30 June 2006

### First-passage and Extreme Value Problems in Random Processes

26 - 30 June 2006

Timetable

Monday 26 June | ||||

08:30-09:50 | Registration | |||

09:50-10:00 | Welcome | |||

10:00-11:00 | Racz, Z (Eotvos University) |
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Scaling functions for finite-size corrections in extreme statistics | Sem 1 | |||

Applications of extreme statistics are hindered by the notoriously slow convergence of the distributions to their limiting forms, thus studies of finite-size corrections are called for. Here we undertake a systematic investigation of the finite-size scaling functions and their universal aspects for the classical extreme statistics of i.i.d. variables, as well as for strongly correlated variables such as occuring in various surface growth problems. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Majumdar, S (Universite Paris-Sud) |
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A nontrivial constant c=0.29795219028 in one and three dimensional random walks | Sem 1 | |||

Two different random walk problems, one in one dimension and the other in three dimensions, seem to share a nontrivial constant whose numerical value is c=0.29795219028.. In the first problem, this constant shows up in the finite size correction to the expected maximum of a discrete-time random walk on a continuous line with unform jump density. In the second problem, this constant appears as the `Milne extrapolation length' in the expression for flux of discrete-time random walkers to a spherical trap in three dimensions. We prove why the same constant appears in the two problems and derive an exact analytical formula for this constant. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:10 | Olla, P | |||

First return time distribution for power law correlated Gaussian processes | Sem 1 | |||

It is known that numerical simulations of power law correlated Gaussian processes lead to 1st return time distributions strongly departing from the exponential regime characteristic of the uncorrelated case. An analytical description of the mechanism leading to the distribution profile is here proposed, taking into full account the long memory of the process. Excellent comparison with numerical data is observed. |
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14:10-14:20 | Oshanin, G | |||

Random processes generated by random permutations. | Sem 1 | |||

In this talk, we overview recent results obtained for an exactly solvable model of a non-markovian random walk generated by random permutations of natural series [1,2,3,...,n]. In this model, a random walker moves on a lattice of integers and its moves to the right and to the left are prescribed the sequence of rises and descents characterizing each given permutation of [n]. We determine exactly the probability distribution function of the end-point of the trajectory, its moments, the probability measure of different excursions, as well as different characteristics showing how scrambled the trajectories are. In addition, we discuss properties of 1d and 2d surfaces associated with random permutations and calculate the distribution function of the number of local extrema. As a by-product, we obtain many novel results on intrinsic features of random permutations (statistics of rises, descents, peaks and throughs). |
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14:30-15:00 | Volkov, S | |||

Recent results for randomly-assigned non-i.i.d. labels on trees | Sem 1 | |||

I would like to talk about some recent results for randomly labelled trees and their relevance to some number-theoretical problems and first-passage percolation. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Comtet, A (Universite Paris Sud) |
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Maximum of a brownian path, fluctuating interfaces and related problems | Sem 1 | |||

We present some functionals of the one dimensional Brownian motion which arise in various statistical physics problems: -maximal fluctuation of a growing interface -traversal time of a potential barrier -search algorithm of the maximum of a simple random walk. All these different cases involve certain functionals of the path and its maximum. We show how to compute these distributions by a path integral approach and discuss the link with probabilistic techniques based on meanders and excursions. |
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16:30-17:30 | Poster Session I | |||

17:30-18:30 | Wine Reception | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Tuesday 27 June | ||||

09:00-10:00 | Bray, A (University of Manchester) |
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First-Passage problems in systems with many degrees of freedom | Sem 1 | |||

Recent progress in the study of first-passage problems in systems with many degrees of freedom will be reviewed, and open questions discussed. The systems considered include coarsening systems, fluctuating interfaces, and reaction-diffusion systems. |
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10:00-11:00 | Godreche, C (CEA Saclay) |
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The statistics of occupation times | Sem 1 | |||

I will present a mini-review of some results on the statistics of occupation times for coarsening systems (spin systems, diffusion equation), or for simpler stochastic models (e.g., renewal processes). |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Burkhardt, T (Temple University) |
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The random acceleration process, with applications to granular matter and polymers | Sem 1 | |||

Some recent results for the equilibrium and non-equilibrium statistics of a particle which is randomly accelerated by Gaussian white noise are reviewed. The cases of motion on the half line x > 0 and motion on the finite line 0 < x < 1 with absorbing, partially absorbing, and inelastic boundary conditions are considered. The results for inelastic reflection at the boundary are of interest in connection with driven granular matter. The equilibrium properties of a semi-flexible polmer chain confined in a narrow cylindrical pore are also determined by the random acceleration process. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Dasgupta, C (Indian Institute of Science) |
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Persistence and survival in equilibrium step fluctuations | Sem 1 | |||

The concepts of persistence and survival are used to characterize the first-passage properties of equilibrium fluctuations of steps on a vicinal surface. Analytic and numerical results are obtained for temporal and spatial persistence and survival probabilities and for the probability of persistent large deviations. These theoretical results are found to be in very good agreement with those obtained from experiments in which time-dependent scanning tunneling microscopy is used to study equilibrium step fluctuations. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Klafter, J (Tel Aviv University) |
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First passage and arrival time densities for Levy Flights | Sem 1 | |||

We discuss a few cases related to the first passage time problem when dominated by Lévy jump length distributions , namely, Lévy flights. These include crossing of a barrier, the semi- infinite interval and the nature of leap-overs. The latter is investigated for both symmetric and one-sided Lévy flights. |
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16:30-17:30 | Poster Session II | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Wednesday 28 June | ||||

09:00-10:00 | Astumian, R (University of Maine) |
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The Unreasonable effectiveness of equilibrium-like theory for interpreting non-equilibrium experiments | Sem 1 | |||

There has been great interest in applying the results of statistical mechanics to single molecule experiements. Recent work has highlighted so-called non-equilibrium work-energy relations and Fluctuation Theorems that take on an equilibrium-like (time independent) form. Here I give a very simple heuristic example where an equilibrium result (the barometric law for colloidal particles) arises from theory describing the {\em thermodynamically} non- equilibrium phenomenon of a single colloidal particle falling through solution due to gravity. This simple description arises from the fact that the particle, even while falling, is in {\em mechanical} equilibrium (gravitational force equal the viscous drag force) at every instant. The results are generalized using Onsager's least dissipation approach for stochastic processes to derive time independent equations that hold for thermodynamically non-equilibrium (and even non-stationary) systems. These equations offer great possibilities for rapid determination of thermodynamic parameters from single molecule experiments. |
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10:00-11:00 | Ben-Naim, E (LANL) |
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Nonequilibrium statistical physics of driven granular gases | Sem 1 | |||

Granular systems dissipate kinetic energy and thus, should be driven to maintain a steady-state. In particular, for granular gases, an energy source should balance the dissipation. This talk will review the different steady-states that characterize driven granular gases. When the system is supplied with energy at all scales, as is the case in most of the vigorous driving experiments, the energy input can be modeled using the standard thermal heat bath. The core of the velocity distribution is then close to a Maxwellian but the tail has a stretched exponential form and it is overpopulated with respect to a Maxwellian. When energy is injected at all energy scales, there is an energy cascade from large velocities to small velocities. In this case, the velocity distribution is characterized by a power-law tail. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Masoliver, J (Universitat de Barcelona) |
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Extreme times in finance | Sem 1 | |||

We analyze the problem of extreme events for financial time series and models. The approach will be different according the nature of the data available. This means that for high-frequency data a micoscopic approach (for which the continuous tuime random walk is a good candidate) is needed; while for lower frequency data one can rely on the traditional approach based on diffusion equations. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-14:10 | Bel, G | |||

Weak ergodicity breaking in the continuous time random walk | Sem 1 | |||

The continuous-time random walk (CTRW) model exhibits a nonergodic phase when the average waiting time diverges. The first passage time probability density function for nonbiased and uniformly biased CTRWs is shown to yields the nonergodic properties of the random walk which show strong deviations from Boltzmann-Gibbs theory. Using numerical simulations we generalize the results for the CTRW in a potential field. We derive the distribution function of occupation times in a bounded region of space which in the ergodic phase recovers the Boltzmann-Gibbs theory, while in the nonergodic phase yields a generalized nonergodic statistical law. |
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14:10-14:20 | Guclu, H | |||

Extreme fluctuations in small-world-coupled autonomous systems with relaxational dynamics | Sem 1 | |||

Synchronization is a fundamental problem in natural and artificial coupled multi-component systems. We investigate to what extent small-world couplings (extending the original local relaxational dynamics through the random links) lead to the suppression of extreme fluctuations in the synchronization landscape of such systems. In the absence of the random links, the steady-state landscape is "rough" (strongly de-synchronized state) and the average and the extreme height fluctuations diverge in the same power-law fashion with the system size (number of nodes). With small-world links present, the average size of the fluctuations becomes finite (synchronized state). For exponential-like noise the extreme heights diverge only logarithmically with the number of nodes, while for power-law noise they diverge in a power-law fashion. The statistics of the extreme heights are governed by the Fisher–Tippett–Gumbel and the Fréchet distribution, respectively. We also study the extreme-value scaling and distributions in scale-free networks. We illustrate our findings through an actual synchronization problem in parallel discrete-event simulations. * http://cnls.lanl.gov/External/people/Hasan_Guclu.php - Homepage * http://www.rpi.edu/~korniss - Homepage * http://cnls.lanl.gov/~guclu - Homepage * http://arxiv.org/abs/cond-mat/0311575 - Preprint |
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14:20-14:30 | Kessler, D | |||

Fluctuation-regularized front propagation up a reaction-rate gradient | Sem 1 | |||

We introduce and study a new class of fronts in finite particle number reaction-diffusion systems, corresponding to propagating up a reaction rate gradient. We show that these systems have no traditional mean-field limit, as the nature of the long-time front solution in the stochastic process differs essentially from that obtained by solving the mean-field deterministic reaction-diffusion equations. Instead, one can incorporate some aspects of the fluctuations via introducing a density cutoff. Using this method, we derive analytic expressions for the front velocity dependence on bulk particle density and show self-consistently why this cutoff approach can get the correct leading-order physics. * http://xxx.arxiv.org/abs/cond-mat/0406336 - Fluctuation-regularized Front Propagation Dynamics * http://xxx.arxiv.org/abs/q-bio.PE/0410015 - Recombination dramatically speeds up evolution of finite populations * http://xxx.arxiv.org/abs/cond-mat/0508128 - Front Propagation Dynamics with Exponentially-Distributed Hopping * http://xxx.arxiv.org/abs/cond-mat/0508663 - Front Propagation up a Reaction Rate Gradient |
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14:30-14:40 | Lythe, G | |||

Noisy kinks and diffusion-limited reaction | Sem 1 | |||

Kinks are examples of ``coherent structures'': clearly identifiable localized features in a noisy, spatially-extended system that can be followed as they move about under the influence of fluctuations. In the Phi4 stochastic partial differential equation, a steady-state mean density is dynamically maintained: kinks and antikinks are nucleated in pairs, follow Brownian paths and annihilate on meeting. Thus the kink-antikink reaction rate is controlled by collisions between diffusing particles. Classical treatment of such problems produces a hierarchy of particle correlation functions without an exact solution. However, it is possible to sidestep this hierarchy and find an exact solution for the mean number of particles per unit length as a function of time. We review an exact method for calculating the mean lifetime of a particles in a simplified model, and an exact rate equation in terms of the correlation function. In addition, the distribution of particle lifetimes is calculated under a ``constant-killing-rate'' approximation that compares favourably with the results of numerical experiments. Related Links * http://maths.leeds.ac.uk/~grant |
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14:40-14:50 | Taitelbaum, H | |||

Interface roughening dynamics of spreading droplets | Sem 1 | |||

We review our recent experimental data of interface roughening dynamics of spreading mercury droplets on thin films (silver or gold), obtained using optical microscopy and other techniques (AFM, SEM). We discuss the various results obtained for the roughness and growth exponents associated with the interface dynamics, and their universality classes. We analyze the temporal width fluctuations, obtained for single interfaces, and show that these fluctuations contain information on the lateral correlation length of these interfaces. We show how to extract this length from experimental data, and demonstrate the validity of this method in a wide range of growing interfaces (droplet spreading experiments as well as water imbibition on paper). Finally, we discuss the persistence exponents of these systems. References: 1. A. Be'er, Y. Lereah and H. Taitelbaum, Physica A, 285, 156 (2000). 2. A. Be'er, Y. Lereah, I. Hecht and H. Taitelbaum, Physica A, 302, 297 (2001). 3. A. Be'er, Y. Lereah, A. Frydman and H. Taitelbaum, Physica A, 314, 325 (2002). 4. A. Be'er and Y. Lereah, J. of Microscopy, 208, 148 (2002). 5. I. Hecht and H. Taitelbaum, Phys. Rev. E, 70, 046307 (2004). 6. A. Be'er, I. Hecht and H. Taitelbaum, Phys. Rev. E, 72, 031606 (2005). 7. I. Hecht, A. Be'er and H. Taitelbaum, Fluctuation and Noise Letters, 5, L319 (2005). |
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15:00-15:30 | Tea | |||

15:30-16:30 | Short talks 2 | |||

20:00-18:00 | Conference Dinner at Robinson College (Dining Hall) |

Thursday 29 June | ||||

09:00-10:00 | Bramson, M (University of Minnesota) |
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Tightness for the minimum displacement of branching random walk and some other old problems | Sem 1 | |||

Study of solutions of certain families of semilinear heat equations dates back to Kolmogorov-Petrovsky-Piscounov in 1937; since then this problem has been thoroughly analyzed. Substantially less is known about the behavior of their discrete time analogs; several basic questions have been unresolved since the 1970's. In the probabilistic context, the continuous time problem corresponds to the minimum displacement of branching Brownian motion, and the discrete time problem to the minimum displacement of branching random walk. Here, we summarize this background and present some new results for branching random walk. |
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10:00-11:00 | Dean, D (Universite Paul Sabatier) |
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Phase transition in the Aldous-Shields Model of growing trees | Sem 1 | |||

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to $c^{-l}$ where $c$ is a positive parameter and $l$ is the distance of the perimeter site from the root. For $c=1$, this model corresponds to random binary search trees and for $c=2$ it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value $c=\sqrt{2}$. While for $c>\sqrt{2}$ the variance is proportional to the mean and the distribution is normal, for $c<\sqrt{2}$ the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at $c=\sqrt{m}$. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Pemantle, R (Pennsylvania) |
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Quantum random walks | Sem 1 | |||

A problem posed by Aldous is to estimate the complexity of finding a (1 - epsilon)-optimal particle in a branching random walk. This is computed in terms of the probability of existence of a trajectory staying forever above the critical drift minus epsilon. (it is known that no particle can stay above the critical drift forever). I will then discuss the computation of this probability, in a continous time (branching Brownian motion) setting, which involves estimating solutions to the KPP equation. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kearney, M (University of Surrey) |
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Exact solutions for first-passage and related problems in certain classes of queueing system | Sem 1 | |||

This talk will examine discrete and continuous time queueing systems in the context of recognising the so-called busy period as the first-passage time of a random walk process. As well as identifying the queue duration (busy-period) distribution, consideration is also given to the distribution of the maximum (extreme) queue length during a busy period and, much harder, the distribution of the total waiting time (area under the curve) during a busy period. Physical examples of interest include traffic jams, Abelian sandpile (avalanche) models in the compact directed percolation universality class, and the statistics of lattice polygon models. Throughout, the emphasis is on providing exact solutions. |
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15:00-15:30 | Tea | |||

15:30-16:30 | ben-Avraham, D (Clarkson University) |
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One-species diffusion-limited reactions on the Bethe lattice | Sem 1 | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Friday 30 June | ||||

09:00-10:00 | Bouchaud, J-P (CEA Saclay) |
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Extreme value problems in random matrix theory, spin glasses and directed polymers | Sem 1 | |||

We will review a few applications of extreme values in the theory of disordered systems and mention several open problems, in particular concerning the generalisation of Parisi's solution or of the Tracy-Widom distribution when the disorder has "fat tails". |
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10:00-11:00 | Benichou, O (College de France) |
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Optimal search strategies for hidden targets | Sem 1 | |||

Many physical, chemical or biological problems can be rephrased as search processes, involving a searcher and a target of unknown position. We show that intermittent search strategies, alternating active search phases and non reactive displacement phases, are universal for a wide class of problems involving search time optimization. More precisely, we address the general question of determining in which cases a searcher should, or should not, interrupt his search activity by "losing" time in non reactive phases of mere displacement, and which durations of each phase optimize the search time. Using a representative analytical model, we show that intermittent strategies do optimize the search time as soon as the target is "difficult" to detect, and we explicitly give the optimal search strategies, which depend on the memory skills of the searcher. |
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11:00-11:30 | Coffee | |||

11:30-11:40 | Bertin, E | |||

Relation between global fluctuations in non-equilibrium systems and Gumbel statistics | Sem 1 | |||

We explain how the statistics of global observables in non-equilibrium or correlated systems can be related to extreme value problems and to Gumbel statistics. This relationship then naturally leads to the emergence of the generalized Gumbel distribution G_a(x), with a real index a, in the study of global fluctuations. To illustrate these results, we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice, with local dissipation, in which the fluctuations of the global energy are precisely described by the generalized Gumbel distribution. Related Links * http://arxiv.org/abs/cond-mat/0506166 - Preprint version of the letter (to appear in PRL) |
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11:40-11:50 | Davidsen, J | |||

Earthquake recurrence as a record breaking process | Sem 1 | |||

Extending the central concept of recurrence times for a point process to recurrent events in space-time allows us to characterize seismicity as a record breaking process using only spatiotemporal relations among events. Linking record breaking events with edges between nodes in a graph generates a complex dynamical network isolated from any length, time or magnitude scales set by the observer. For Southern California, the network of recurrences reveals new statistical features of seismicity with robust scaling laws. The rupture length and its scaling with magnitude emerges as a generic measure for distance between recurrent events. Further, the relative separations for subsequent records in space (or time) form a hierarchy with unexpected scaling properties. Related Links * http://www.pks.mpg.de/~davidsen/ - homepage |
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12:00-12:10 | Mallick, K | |||

Effects of the low frequencies of noise on On--Off intermittency | Sem 1 | |||

A bifurcating system subject to multiplicative noise can exhibit on--off intermittency close to the instability threshold. For a canonical system, we discuss the dependence of this intermittency on the Power Spectrum Density (PSD) of the noise. Our study is based on the calculation of the Probability Density Function (PDF) of the unstable variable. We derive analytical results for some particular types of noises and interpret them in the framework of on-off intermittency. Besides, we perform a cumulant expansion \cite{VanKampen1} for a random noise with arbitrary power spectrum density and show that the intermittent regime is controlled by the ratio between the departure from the threshold and the value of the PSD of the noise at zero frequency. Our results are in agreement with numerical simulations performed with two types of random perturbations: colored Gaussian noise and deterministic fluctuations of a chaotic variable. Extensions of this study to another, more complex, system are presented and the underlying mechanisms are discussed. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Redner, S (Boston University) |
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On the role of global warming on the statistics of record-breaking temperatures | Sem 1 | |||

We theoretically study long-term statistics of record-breaking daily temperatures and validate these predictions by simulations, and by comparing with 126 years of daily temperature data from the city of Philadelphia. Using extreme statistics, we derive the number and the magnitude of record temperature events, based on the observed Gaussian daily temperature distribution in Philadelphia. We then consider the effect of global warming on record temperature events. The current warming rate is insufficient to measurably influence the frequency of these events over the time range of the available observations. |
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15:00-15:30 | Tea | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |