Probabilistic nilpotence in infinite groups

Let W be an integer-valued random variable satisfying \(E[W] =: \delta \ge 0\) and \(P(W 0\), and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer \(x\in \mathbb Z\), the size of the next step is an independent random variable with the same distribution as W. We show that this self-interacting random walk is recurrent if \(\delta \le 1\) and transient if \(\delta >1\). This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.

We present here the main result from [8] and explain how to use Kashiwara crystal basis theory to associate a random walk to each minuscule irreducible representation V of a simple Lie algebra; the generalized Pitman transform defined in [10] for similar random walks with uniform distributions yields yet a Markov chain when the crystal attached to V is endowed with a probability distribution compatible with its weight graduation. The main probabilistic argument in our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice [8]. We present some explicit examples, which can be computed using insertion schemes on tableaux described in [9].

Automorphism groups of nilpotent groups and spaces

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

Sampling from large graphs is an area which is of great interest, particularly with the recent emergence of huge structures such as Online Social Networks. These often contain hundreds of millions of vertices and billions of edges. The large size of these networks makes it computationally expensive to obtain structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer a problem.In this paper we develop an analysis of random walks, a commonly used method of sampling from networks. We present a method of biassing the random walk to acquire a complete sample of high degree vertices of social networks, or similar graphs. The preferential attachment model is a common method to generate graphs with a power law degree sequence. For this model, we prove that this sampling method is successful with high probability. We also make experimental studies of the method on various real world networks.For t-vertex graphs G(t) generated by a preferential attachment process, we analyze a biassed random walk which makes transitions along undirected edges {x,y} proportional to [d(x)d(y)]b, where d(x) is the degree of vertex x and b > 0 is a constant parameter. Let S(a) be the set of all vertices of degree at least ta in G(t). We show that for some b approx 2/3, if the biassed random walk starts at an arbitrary vertex of S(a), then with high probability the set S(a) can be discovered completely in ~O(t1-(4/3)a+d) steps, where d is a very small positive constant. The notation ~O ignores poly-log t factors.The preferential attachment process generates graphs with power law 3, so the above example is a special case of this result. For graphs with degree sequence power law c>2 generated by a generalized preferential attachment process, a random walk with transitions along undirected edges {x,y} proportional to (d(x)d(y))(c-2)/2, discovers the set S(a) completely in ~O(t1-a(c-2)+d) steps with high probability. The cover time of the graph is ~O(t).Our results say that if we search preferential attachment graphs with a bias b=(c-2)/2 proportional to the power law c then, (i) we can find all high degree vertices quickly, and (ii) the time to discover all vertices is not much higher than in the case of a simple random walk. We conduct experimental tests on generated networks and real-world networks, which confirm these two properties.

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

The degradation suffered when pulses satisfying the Nyquist criterion are used to transmit binary data at a rate faster than the Nyquist rate over the ideal band-limited (brick-wall) channel is studied. The minimum distance between received signals is used as a performance criterion. It is well-known that when Nyquist pulses (i.e. pulses satisfying the Nyquist criterion) are sent at the Nyquist rate, the minimum distance between signal points is the same as the pulse energy. The main result is to show that the minimum distance between received signals is the same as the pulse energy for rates of transmission about 25% beyond the Nyquist rate, which is the best possible result. In fact, one can even identify the precise error event and signaling rate that causes the minimum distance to be no longer equal to the pulse energy. The mathematical formulation of the problem is analyzed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

Consider a random walk $S_{i}=\xi_{1}+\cdots+\xi_{i}$, $i\in\mathbb{N}$, whose increments $\xi_{1},\xi_{2},\ldots$ are independent identically distributed random vectors in $\mathbb{R}^{d}$ such that $\xi_{1}$ has the same law as $-\xi_{1}$ and $\mathbb{P}[\xi_{1}\in H]=0$ for every affine hyperplane $H\subset\mathbb{R}^{d}$. Our main result is the distribution-free formula \[\mathbb{E}\bigg[\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\mathbb{1}_{\{0\notin\operatorname{Conv}(S_{i_{1}},\ldots,S_{i_{k}})\}}\bigg]=2\binom{n}{k}\frac{B(k,d-1)+B(k,d-3)+\cdots}{2^{k}k!},\] where the $B(k,j)$’s are defined by their generating function $(t+1)(t+3)\ldots(t+2k-1)=\sum_{j=0}^{k}B(k,j)t^{j}$. The expected number of $k$-tuples above admits the following geometric interpretation: it is the expected number of $k$-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type $B_{n}$ that are not intersected by a generic linear subspace $L\subset\mathbb{R}^{n}$ of codimension $d$. The case $d=1$ turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required.

The scaling limits for d-dimensional random walks perturbed by an attractive force toward the origin are studied under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. Our results extend those obtained by (2) from the mean-zero Gaussian to non-Gaussian setting under the absence of the wall. 1. Introduction and main result. It is a general principle in the study of various kinds of scaling limits that the limit points, at least at the level of law of large numbers, can be characterized by variational problems which minimize the rate functionals of the corresponding large deviation principles. However, if the rate functional admits several minimizers, the large deviation principle is not sucient to give an appropriate answer. This paper discusses such problem, especially for random walks on R d perturbed by an attractive force toward the origin 0 2 R d , motivated by certain models for interfaces or directed polymers.

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami (2001), essentially continues to hold for a general real symmetric Wigner matrix. To obtain this result, we adapt the approach by Götze and Kösters (2008), who proved the analogous result for the Hermitian case.

The evaluation of fourier transforms by the sampling method

Let $r(n)$ be the radius of the largest disc covered by $S(1),\ldots, S(n)$, where $\{S(k); k = 1, 2,\ldots\}$ is the simple symmetric random walk on $Z^2$. The main result tells us that $r(n) \geq n^{1/50}$ a.s. for all but finitely many $n$.

Let W W be an infinite irreducible Coxeter group with ( s 1 , … , s n ) (s_1, \ldots , s_n) the simple generators. We give a short proof that the word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s n ⋯ s 1 s 2 ⋯ s n s_n \cdots s_1 s_2 \cdots s_n is reduced for any number of repetitions of s 1 s 2 ⋯ s n s_1 s_2 \cdots s_n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 w_0 which is obtained from the semi-infinite word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s n ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots by interchanging commuting elements and taking a prefix.