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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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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.
A CLASS OF WEAKLY SELFAVOIDING WALKS
"... We define a class of weakly selfavoiding walks on the integers by conditioning a simple random walk of length n to have a pfold selfintersection local time smaller than n β, where 1 < β < (p+1)/2. We show that the conditioned paths grow of order n α, where α = (p − β)/(p − 1), and also pr ..."
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We define a class of weakly selfavoiding walks on the integers by conditioning a simple random walk of length n to have a pfold selfintersection local time smaller than n β, where 1 < β < (p+1)/2. We show that the conditioned paths grow of order n α, where α = (p − β)/(p − 1), and also prove a coarse large deviation principle for the order of growth.
A double phase transition arising from Brownian entropic repulsion
, 2008
"... Abstract. We analyze onedimensional Brownian motion conditioned on a selfrepelling behaviour. In the main result of this paper, it is shown that a double phase transition occurs when the growth of the local time at the origin is constrained (in a suitable way) to be slower than the function f(t) ..."
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Abstract. We analyze onedimensional Brownian motion conditioned on a selfrepelling behaviour. In the main result of this paper, it is shown that a double phase transition occurs when the growth of the local time at the origin is constrained (in a suitable way) to be slower than the function f(t) = √ t(log t) −c at every time. In the subcritical phase (c < 0), the process is recurrent and the local time at 0 is diffusive. In the intermediary phase (0 < c ≤ 1), the process is recurrent but the local time grows much slower than the constraint f. Finally in the supercritical phase (c> 1), the process becomes transient. The proof exploits the Brownian entropic repulsion phenomenon.
Brownian entropic repulsion
, 2008
"... Abstract. We consider onedimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptot ..."
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Abstract. We consider onedimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process is asymptotically ballistic but with an unknown speed.