Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that, almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values go to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to [0, 1] ⊂ ℂ (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0, 1/2]. In addition we prove that the largest angle converges to π in probability. Our approach is probabilistic, and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0, 1/2]. The stationary distribution of this limit chain is particularly important in our study.

We study the diameter of 1, the largest component of the Erdős–Rényi random graph (n, p) in the emerging supercritical phase, i.e., for p = where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of .

We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of 1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of 1 is w.h.p. asymptotic to , and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to .

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from 26 April to 2 May. This meeting focused on the common themes of Combinatorics, Discrete Probability and Theoretical Computer Science, and the lectures, many of which were given by young participants, stimulated fruitful discussions. The open problems session held during the meeting, and the fact that the participants work in different and related topics, encouraged interesting discussions and collaborations.

In this paper we study the diameter of the random graph G(n, p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ > 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish two-point concentration in the case that np → ∞. For p =(1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n, p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.

Ordering constraints are formally analogous to instances of the satisfiability problem in conjunctive normal form, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering if and only if the relative ordering of its variables obeys the constraint considered.

The naturally arising satisfiability problems are NP-complete for many types of constraints. We look at random ordering constraints. Previous work of the author shows that there is a sharp unsatisfiability threshold for certain types of constraints. The value of the threshold, however, is essentially undetermined. We pursue the problem of approximating the precise value of the threshold. We show that random instances of the betweenness constraint are satisfiable with high probability if the number of randomly picked clauses is ≤0.92n, where n is the number of variables considered. This improves the previous bound, which is <0.82n random clauses. The proof is based on a binary relaxation of the betweenness constraint and involves some ideas not used before in the area of random ordering constraints.

The properties characterizing Sturmian words are considered forwords on multiliteral alphabets. Wesummarize various generalizations of Sturmian words tomultiliteral alphabets and enlarge the list of knownrelationships among these generalizations.We provide a new equivalent definition of rich wordsand make use of it in the study of generalizations of Sturmian words based on palindromes.We also collect many examples of infinite words to illustrate differences in thegeneralized definitions of Sturmian words.

The paper deals with some classes of two-dimensional recognizablelanguages of “high complexity”,in a sense specified in the paper and motivated by some necessary conditions holding for recognizable andunambiguous languages. For such classes we can solve someopen questions related to unambiguity, finite ambiguityand complementation. Then we reformulate a necessary condition for recognizability stated by Matz, introducing a new complexity function.We solve an open question proposed by Matz, showing that all the known necessary conditions for recognizability of a language and its complement arenot sufficient. The proof relies on a family of languages defined by functions.

Quantum Coherent Spaces were introduced by Girard as a quantum framework where to interpret the exponential-free fragment of Linear Logic. Aim of this paper is to extend Girard's interpretation to a subsystem of linear logic with bounded exponentials. We provide deduction rules for the bounded exponentials and, correspondingly, we introduce the novel notion of bounded exponentials of Quantum Coherent Spaces. We show that the latter provide a categorical model of our system. In order to do that, we first study properties of the category of Quantum Coherent Spaces.

We study the relation between the standard two-way automata andmore powerful devices, namely, two-way finite automata equippedwith some $\ell$ additional “pebbles” that are movable alongthe input tape, but their use is restricted (nested) ina stack-like fashion. Similarly as in the case of the classicaltwo-way machines, it is not known whether there existsa polynomial trade-off, in the number of states, between thenondeterministic and deterministic two-way automata with $\ell$ nested pebbles. However, we show that these two machine modelsare not independent: if there exists a polynomial trade-off forthe classical two-way automata, then, for each $\ell$ ≥ 0,there must also exist a polynomial trade-off for the two-wayautomata with $\ell$ nested pebbles. Thus, we have an upwardcollapse (or a downward separation) from the classical two-wayautomata to more powerful pebble automata, still staying withinthe class of regular languages. The same upward collapse holdsfor complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machinecan be, by using suitable inputs, simulated by a classicaltwo-way automaton (and vice versa), with only a linear number ofstates, despite the existing exponential blow-up between theclassical and pebble two-way machines.

We consider directed figures defined as labelled polyominoes with designated start andend points, with two types of catenation operations. We are especially interested in codicityverification for sets of figures, and we show that depending on the catenation type the questionwhether a given set of directed figures is a code is decidable or not. In the former case we give aconstructive proof which leads to a straightforward algorithm.

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model $\mbox{IPM}_k$ (n),a generalization of the sand pile model $\mbox{SPM}$ (n).More precisely, for any fixed integer k, we show thatthe negative lexicographic ordering naturally identifies a tree structure on the lattice $\mbox{IPM}_k$ (n):this lets us design an algorithm which generates all the ice piles of $\mbox{IPM}_k$ (n)in amortized timeO(1)and in spaceO( $\sqrt n$ ).

We introduce a polynomial invariant of graphs on surfaces, PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for PG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, PG specializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomial PG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

Let Ωn be the nn-element set consisting of all functions that have {1, 2, 3, . . ., n} as both domain and codomain. Let T(f) be the order of f, i.e., the period of the sequence f, f(2), f(3), f(4) . . . of compositional iterates. A closely related number, B(f) = the product of the lengths of the cycles of f, has previously been used as an approximation for T. This paper proves that the average values of these two quantities are quite different. The expected value of T iswhere k0 is a complicated but explicitly defined constant that is approximately 3.36. The expected value of B is much larger: