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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Betacoalescents and continuous stable random trees
, 2006
"... Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case t ..."
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Cited by 47 (15 self)
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Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta(2 − α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here we use a recent result of Birkner et al. to prove that Betacoalescents can be embedded in continuous stable random trees, about which much is known due to recent progress of Duquesne and Le Gall. Our proof is based on a construction of the DonnellyKurtz lookdown process using continuous random trees which is of independent interest. This produces a number of results concerning the smalltime behavior of Betacoalescents. Most notably, we recover an almost sure limit theorem of the authors for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and allele frequency spectrum associated with mutations in the context of population genetics.
A random environment for linearly edgereinforced random walks on infinite graphs. Probab. Theory Related Fields 138
, 2007
"... We consider linearly edgereinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a random walk in a timeindependent random environment given by strictly positive weights on the edges. Furthermore, we prove bounds for the rand ..."
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Cited by 15 (4 self)
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We consider linearly edgereinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a random walk in a timeindependent random environment given by strictly positive weights on the edges. Furthermore, we prove bounds for the random environment, uniform, among others, in the size of the graph. 34 1
Maximally stable Gaussian partitions with discrete applications
 Israel J. Math
"... Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat k ..."
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Cited by 13 (3 self)
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Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications: • An optimality result for majority in the context of Condorcet voting. • A proof of a conjecture on “cosmic coin tossing ” for low influence functions. We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the “Double Bubble ” theorem and show that it implies: • A proof of the “Plurality is Stablest Conjecture”. • That the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture.
The Random Average Process and Random Walk in a SpaceTime Random Environment in One Dimension
 MATHEMATICAL PHYSICS
, 2006
"... We study spacetime fluctuations around a characteristic line for a onedimensional interacting system known as the random average process. The state of this system is a realvalued function on the integers. New values of the function are created by averaging previous values with random weights. Th ..."
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Cited by 11 (4 self)
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We study spacetime fluctuations around a characteristic line for a onedimensional interacting system known as the random average process. The state of this system is a realvalued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n 1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known productform invariant distributions, this limit is a twoparameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a spacetime random environment. These limits also happen at scale n 1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.
Fractional martingales and characterization of the fractional Brownian motion
 Ann. Probab
, 2009
"... In this paper we introduce the notion of αmartingale as the fractional derivative of order α of a continuous local martingale, where α ∈ (−1 1 2), and we show that it has a nonzero finite variation of 2 order ..."
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Cited by 9 (0 self)
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In this paper we introduce the notion of αmartingale as the fractional derivative of order α of a continuous local martingale, where α ∈ (−1 1 2), and we show that it has a nonzero finite variation of 2 order
A Nerode, Effective dimension of points visited by Brownian motion, Theoretical Computer Science 410
, 2009
"... We consider the individual points on a MartinLöf random path of Brownian motion. We show (1) that Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension < 1. The proof of (1) shows ..."
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Cited by 9 (0 self)
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We consider the individual points on a MartinLöf random path of Brownian motion. We show (1) that Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension < 1. The proof of (1) shows that for almost all times t, the path f is MartinLöf random relative to t and so the effective dimension of (t, f(t)) is 2. Keywords: Brownian motion, algorithmic randomness, effective randomness 1
Ballistic random walk in a random environment with a forbidden direction
, 2005
"... We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. Homogenization and regeneration techniques combine to prove a law of large numbers and an averaged invariance principle. The assumptions are nonnestling and 1 + ε (resp. ..."
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Cited by 8 (5 self)
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We consider a ballistic random walk in an i.i.d. random environment that does not allow retreating in a certain fixed direction. Homogenization and regeneration techniques combine to prove a law of large numbers and an averaged invariance principle. The assumptions are nonnestling and 1 + ε (resp. 2 + ε) moments for the step of the walk uniformly in the environment, for the law of large numbers (resp. invariance principles). We also investigate invariance principles under fixed environments, and invariance principles for the environmentdependent mean of the walk.
New Maximally Stable Gaussian Partitions with Discrete Applications
, 2009
"... Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures ..."
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Cited by 4 (1 self)
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Gaussian noise stability results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice. We propose two Gaussian noise stability conjectures and derive consequences of the conjectures in hardness of approximation and social choice. Both conjectures generalize isoperimetric results by Borell on the heat kernel. One of the conjectures may be also be viewed as a generalization of the ”Double Bubble ” theorem. The applications of the conjectures include an optimality result for majority in the context of Condorcet voting and a proof that the FriezeJerrum SDP for MAXqCUT achieves the optimal approximation factor assuming the Unique Games Conjecture. We finally derive a short proof of the first conjecture based on the extended Riesz inequality. 1