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12
Nonbacktracking random walks and cogrowth of graphs
 Canadian Journal of Mathematics
"... Abstract. Let X be a locally finite, connected graph without vertices of degree 1. Nonbacktracking random walk moves at each step with equal probability to one of the “forward ” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a ..."
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Abstract. Let X be a locally finite, connected graph without vertices of degree 1. Nonbacktracking random walk moves at each step with equal probability to one of the “forward ” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of X. Thus we obtain for infinite X that the nstep nonbacktracking transition probabilities tend to zero, and we can also compute their limit when X is finite. This provides a short proof of an old result concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when X is nonregular, but small cycles are dense in X, we show that the graph X is nonamenable if and only if the nonbacktracking nstep transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen. 1
Multiplicative measures on free groups
 INT. J. ALGEBRA COMP
, 2002
"... 1 How one can measure subsets in the free group? 1.1 Motivation The present paper is motivated by needs of practical computations in finitely presented groups. In particular, we wish to develop tools which can be used in the analysis of the “practical ” complexity of algorithmic problems for discret ..."
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1 How one can measure subsets in the free group? 1.1 Motivation The present paper is motivated by needs of practical computations in finitely presented groups. In particular, we wish to develop tools which can be used in the analysis of the “practical ” complexity of algorithmic problems for discrete infinite groups, as well as in the analysis of the behaviour of heuristic (e.g. genetic) algorithms for infinite groups [22, 23]. In most computerbased computations in finitely presented groups G = F/R the elements are represented as freely reduced words in the free group F, with procedures for comparing their images in the factor group G = F/R. Therefore the ambient algebraic structure in all our considerations is the free group F = F(X) on a finite set X = {x1,...,xm}. We identify F with the set of all freely reduced words in the alphabet X ∪X −1, with the multiplication given by concatenation of words with the subsequent free reduction. The most natural and convenient way to generate pseudorandom elements
Counting paths in graphs
 Ensignment Math
, 1999
"... Abstract. We give a simple combinatorial proof of a formula that extends a result by Grigorchuk [Gri78a, Gri78b] (rediscovered by Cohen [Coh82]) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of ‘bumps ’ on paths in a graph: in a ..."
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Abstract. We give a simple combinatorial proof of a formula that extends a result by Grigorchuk [Gri78a, Gri78b] (rediscovered by Cohen [Coh82]) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of ‘bumps ’ on paths in a graph: in a dregular (not necessarily transitive) nonoriented graph let the series G(t) count all paths between two fixed points weighted by their length tlength, and F(u, t) count the same paths, weighted as unumber of bumpstlength. Then one has
Expansion of random graphs: New proofs, new results
, 2014
"... We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let Γ be a random dregular graph on n vertices, and let λ be the largest absolut ..."
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Cited by 10 (0 self)
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We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let Γ be a random dregular graph on n vertices, and let λ be the largest absolute value of a nontrivial eigenvalue of its adjacency matrix. It was conjectured by Alon [Alo86] that a random dregular graph is “almost Ramanujan”, in the following sense: for every ε> 0, a.a.s. λ < 2 d − 1 + ε. Friedman famously presented a proof of this conjecture in [Fri08]. Here we suggest a new, substantially simpler proof of a nearlyoptimal result: we show that a random dregular graph satisfies λ < 2 d − 1 + 1 asymptotically almost surely. A main advantage of our approach is that it is applicable to a generalized conjecture: A dregular graph on n vertices is an ncovering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph Ω, we study the spectrum of Γ, a random ncovering of Ω. Let
Quotient tests and random walks in computational group theory
, 2005
"... For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few sim ..."
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Cited by 3 (1 self)
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For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few simple case studies of quotient tests for the classical identity, word, conjugacy problems in groups. We attempt to combine a rigorous analytic study with the assessment of algorithms from the practical point of view. It appears that, in case of finite quotient groups G/K, the efficiency of the quotient test very much depends on the mixing times for random walks on the Cayley graph of G/K.
Growth series and random walks on some hyperbolic graphs
 in print, Monatsh. Math
, 2002
"... Abstract. Consider the tesselation of the hyperbolic plane by mgons, ℓ per vertex. In its 1skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of reduced loops in these graphs. We then app ..."
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Cited by 2 (1 self)
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Abstract. Consider the tesselation of the hyperbolic plane by mgons, ℓ per vertex. In its 1skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate holly trees, a family of reduced loops in these graphs. We then apply Grigorchuk’s result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs. 1.
Cogrowth of Arbitrary Graphs
"... Abstract. A “cogrowth set ” of a graph G is the set of vertices in the universal cover of G which are mapped by the universal covering map onto a given vertex of G. Roughly speaking, a cogrowth set is large if and only if G is small. In particular, when G is regular, a cogrowth constant (a measure o ..."
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Abstract. A “cogrowth set ” of a graph G is the set of vertices in the universal cover of G which are mapped by the universal covering map onto a given vertex of G. Roughly speaking, a cogrowth set is large if and only if G is small. In particular, when G is regular, a cogrowth constant (a measure of the size of the cogrowth set) exists and has been shown to be as large as possible if and only if G is amenable. We present two approaches to the problem of extending this to the nonregular case. First, we show that the result above extends to the case when G is not regular but is the cover of a finite graph. This proof is based on some properties of a family of Laplacians related to the zeta function of the covered graph. An example is given where this result fails when G does not cover a finite graph. Second, for any graph with transient covering tree, we define a new cogrowth constant expressed in terms of harmonic measure and show that G is amenable if and only if this constant is 1. Finally, we show that if G covers a finite graph, then the radial limit set of a cogrowth set has largest possible Hausdorff dimension if and only if G is amenable. 1.