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22
Composition of random transpositions
- Department of Mathematics University of British Columbia
, 2005
"... Let Y = (y1,y2,...), y1 ≥ y2 ≥ · · · , be the list of sizes of the cycles in the composition of cn transpositions on the set {1,2,...,n}. We prove that if c> 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where t ..."
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Cited by 23 (1 self)
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Let Y = (y1,y2,...), y1 ≥ y2 ≥ · · · , be the list of sizes of the cycles in the composition of cn transpositions on the set {1,2,...,n}. We prove that if c> 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function f is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process. 1
Emergence of giant cycles and slowdown transition in random transpositions and k-cycles
- Electr. J. Probab
"... Abstract Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the random walk slows down abruptly: the acceleration ..."
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Abstract Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the random walk slows down abruptly: the acceleration (i.e., the second time derivative of the distance) drops from 0 to −∞ at this time as n → ∞. On the other hand, the largest cycle size changes from microscopic to giant. The proof of this last result is considerably simpler and holds more generally than in a previous result of Oded Schramm [19] for random transpositions. It turns out that in the case of random k-cycles, this critical time is proportional to 1/[k(k − 1)], whereas the mixing time is known to be proportional to 1/k.
Mixing times for random k-cycles and coalescence-fragmentation chains
, 1961
"... Dedicated to the memory of ODED SCHRAMM Let Sn be the permutation group on n elements, and consider a random walk on Sn whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proo ..."
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Cited by 8 (1 self)
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Dedicated to the memory of ODED SCHRAMM Let Sn be the permutation group on n elements, and consider a random walk on Sn whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of Sn.
The hyperbolic geometry of random transpositions
- Ann. Probab
, 2005
"... Turn the set of permutations of n objects into a graph Gn by connecting two permutations that differ by one transposition, and let σt be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17–26] showed that the limiting behavior of the ..."
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Cited by 7 (5 self)
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Turn the set of permutations of n objects into a graph Gn by connecting two permutations that differ by one transposition, and let σt be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17–26] showed that the limiting behavior of the distance from the identity at time cn/2 has a phase transition at c = 1. Here we investigate some consequences of this result for the geometry of Gn. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occurs at a critical radius equal to n/4. Let T be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius an for a constant 0 < a < 1. Then when a < 1/4, if the geodesics are suitably chosen, with high probability T is δ-thin for some δ> 0, whereas it is always O(n)-thick when a> 1/4. We also show that the hitting distribution of the sphere of radius an is asymptotically singular with respect to the uniform distribution. Finally, we prove that the critical behavior of this Gromov-like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius an. However, in this case, the critical radius is a = 1 − log2.
Distribution of Segment Lengths in Genome Rearrangements
"... The study of gene orders for constructing phylogenetic trees was introduced by Dobzhansky and Sturtevant in 1938. Different genomes may have homologous genes arranged in different orders. In the early 1990s, Sankoff and colleagues modelled this as ordinary (unsigned) permutations on a set of numbere ..."
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The study of gene orders for constructing phylogenetic trees was introduced by Dobzhansky and Sturtevant in 1938. Different genomes may have homologous genes arranged in different orders. In the early 1990s, Sankoff and colleagues modelled this as ordinary (unsigned) permutations on a set of numbered genes 1, 2,..., n, with biological events such as inversions modelled as operations on the permutations. Signed permutations may be used when the relative strands of the genes are known, and “circular permutations ” may be used for circular genomes. We use combinatorial methods (generating functions, commutative and noncommutative formal power series, asymptotics, recursions, and enumeration formulas) to study the distributions of the number and lengths of conserved segments of genes between two or more unichromosomal genomes, including signed and unsigned genomes, and linear and circular genomes. This generalizes classical work on permutations from the 1940s– 60s by Wolfowitz, Kaplansky, Riordan, Abramson, and Moser, who studied decompositions of permutations into strips of ascending or descending consecutive numbers. In our setting, their work corresponds to comparison of two unsigned genomes (known gene orders, unknown gene orientations). Maple software implementing our formulas is available at
THE EXPECTED NUMBER OF INVERSIONS AFTER n ADJACENT TRANSPOSITIONS
, 2010
"... We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group Sm+1. We then derive from this expression the asymptotic behaviour of this number when n ≡ nm scales with m in various ways. Our starting point is an equivalence ..."
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We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group Sm+1. We then derive from this expression the asymptotic behaviour of this number when n ≡ nm scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.
Computing Genomic Midpoints
, 2005
"... This paper proposes a new algorithm for the genomic median problem that combines greedy and stochastic search. Our computational experiments suggest that for more complex problems our algorithm finds better solutions than previous approaches. In particular we find an improved midpoint for a human-mo ..."
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This paper proposes a new algorithm for the genomic median problem that combines greedy and stochastic search. Our computational experiments suggest that for more complex problems our algorithm finds better solutions than previous approaches. In particular we find an improved midpoint for a human-mouse-rat comparison with 424 markers. In order to understand why such problems are hard, we explore a phase transition in the complexity of the median problem for random data, associated with the emergence of a giant component in the breakpoint graph.
