invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

the trace of spatial brownian motion is capacity equivalent to the unit square we show that with probability 1 the trace b 0 1 of brownian motion in space has positive capacity with respect to exactly the same kernels as the unit square more precisely the energy of occupation measure on b 0 1 in the kernel f x y is bounded above and below by constant multiples of the energy of lebesgue measure on the unit square the constants are random but do not depend on the kernel as an application we give almost sure asymptotics for the probability that an stable process approaches within of b 0 1 conditional on b 0 1

asymptotic expansions of witten reshetikhin turaev invariants for some simple 3 manifolds the link l this paper investigates the invariant z m 0 when g sk for a simple family of rational homology 3 spheres obtained by integer surgery around 2 n type torus knots in particular we find a closed formula for a formal power series z m eq h in h q 1 from which zf m 0 may be derived for all sufficiently large primes r we show that this formal power series may be viewed as the asymptotic expansion around 4 1 of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition for these particular manifolds most of which are not z homology spheres this extends work of ohtsuki and murakami in which the existence of power series with rational coefficients related to z m 0 was demonstrated for rational homology spheres the coefficients in the formal power series z m are expected to be identical to those obtained from a perturbative expansion of the witten chem simons path integral formula for z m 0 0 199 5 american institute of physics

the design of trellis coded mpsk for fading channels set partitioning for optimum code design a previous work on criteria for designing trellis coded mpsk modulation to achieve minimum error probability performance on the rician fading channel see ibid vol 36 no 9 p 1004 1012 sep 1988 is extended it is demonstrated that allowing for multiple symbols per trellis branch i e multiple trellis coded modulation mtcm provides an additional degree of freedom for designing a code to meet the optimization on the fading channel diversities larger than those achievable with conventional trellis codes having the same number of trellis states are now attainable it is under these conditions that mtcm achieves its full potential

the alexander orbach conjecture holds in high dimensions we examine the incipient infinite cluster iic of critical percolation in regimes where mean field behavior has been established namely when the dimension d is large enough or when d 6 and the lattice is sufficiently spread out we find that random walk on the iic exhibits anomalous diffusion with the spectral dimension d s frac 4 3 that is p t x x t 2 3 o 1 this establishes a conjecture of alexander and orbach j phys lett paris 43 625 631 1982 en route we calculate the one arm exponent with respect to the intrinsic distance

the continuum limit of critical random graphs we consider the erd s renyi random graph g n p inside the critical window that is when p 1 n n 4 3 for some fixed lambda in mathbb r we prove that the sequence of connected components of g n p considered as metric spaces using the graph distance rescaled by n 1 3 converges towards a sequence of continuous compact metric spaces the result relies on a bijection between graphs and certain marked random walks and the theory of continuum random trees our result gives access to the answers to a great many questions about distances in critical random graphs in particular we deduce that the diameter of g n p rescaled by n 1 3 converges in distribution to an absolutely continuous random variable with finite mean

large deviations for random walks on galton watson trees averaging and uncertainty in the study of large deviations for random walks in random environment a key distinction has emerged between quenched asymptotics conditional on the environment and annealed asymptotics obtained from averaging over environments in this paper we consider a simple random walk x n on a galton watson tree t i e on the family tree arising from a supercritical branching process denote by x n the distance between the node x n and the root of t our main result is the almost sure equality of the large deviation rate function for x n n under the quenched measure conditional upon t and the rate function for the same ratio under the annealed measure averaging on t according to the galton watson distribution this equality hinges on a concentration of measure phenomenon for the momentum of the walk the momentum at level n for a specific tree t is the average over random walk paths of the forward drift at the hitting point of that level this concentration or certainty is a consequence of the uncertainty in the location of the hitting point we also obtain similar results when x n is a biased walk on a galton watson tree even though in that case there is no known formula for the asymptotic speed our arguments rely at several points on a ubiquity lemma for galton watson trees due to grimmett and kesten 1984

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

construction of new mpsk trellis codes for fading channels in this paper a new methodology for design of mpsk trellis codes on fading channels is proposed the proposed design criteria based on effective code length the minimum product distance and the minimum euclidean distance are used to construct new 8 psk and 16 psk trellis codes with improved bit error rate performance on fading channels the code performance is evaluated analytically and by monte carlo simulation the results show that the proposed codes achieve significant improvements relative to the best known codes

some block and trellis coded modulations for the rayleigh fading channel the error performance of a modulation code over a channel depends on several distance parameters and the path multiplicity of the code for the awgn channel the error performance of a modulation code depends mainly on its minimum squared euclidean distance and path multiplicity for the rayleigh fading channel however the error performance of a modulation code depends strongly on its minimum symbol distance minimum product distance and path multiplicity it depends on the minimum squared euclidean distance in a lesser degree this paper is concerned with the construction of block and trellis mpsk modulation codes for the rayleigh fading channel in each construction the distance parameters are chosen to achieve good error performance with reduced decoding complexity

adaptive coded modulation for fading channels we apply coset codes to adaptive modulation in fading channels adaptive modulation is a powerful technique to improve the energy efficiency and increase the data rate over a fading channel coset codes are a natural choice to use with adaptive modulation since the channel coding and modulation designs are separable therefore trellis and lattice codes designed for additive white gaussian noise awgn channels can be superimposed on adaptive modulation for fading channels with the same approximate coding gains we first describe the methodology for combining coset codes with a general class of adaptive modulation techniques we then apply this methodology to a spectrally efficient adaptive m ary quadrature amplitude modulation mqam to obtain trellis coded adaptive mqam we present analytical and simulation results for this design which show an effective coding gain of 3 db relative to uncoded adaptive mqam for a simple four state trellis code and an effective 3 6 db coding gain for an eight state trellis code more complex trellis codes are shown to achieve higher gains we also compare the performance of trellis coded adaptive mqam to that of coded modulation with built in time diversity and fixed rate modulation the adaptive method exhibits a power savings of up to 20 db

critical percolation clusters in seven dimensions and on a complete graph we study critical bond percolation on a seven dimensional hypercubic lattice with periodic boundary conditions 7d and on the complete graph cg of finite volume number of vertices v we numerically confirm that for both cases the critical number density n s v of clusters of size s obeys a scaling form n s v s n over s v d f with identical volume fractal dimension d f 2 3 and exponent 1 1 d f 5 2 we then classify occupied bonds into bridge bonds which includes branch and junction bonds and nonbridge bonds a bridge bond is a branch bond if and only if its deletion produces at least one tree deleting branch bonds from percolation configurations produces leaf free configurations whereas deleting all bridge bonds leads to bridge free configurations composed of blobs it is shown that the fraction of nonbridge biconnected bonds vanishes n cg 0 for large cgs but converges to a finite value n 7d 0 0061931 7 for the 7d hypercube further we observe that while the bridge free dimension d bf 1 3 holds for both the cg and 7d cases the volume fractal dimensions of the leaf free clusters are different d lf 7d 0 669 9 2 3 and d lf cg 0 3337 17 1 3 on the cg and in 7d the whole leaf free and bridge free clusters all have the shortest path volume fractal dimension d min 1 3 characterizing their graph diameters we also study the behavior of the number and the size distribution of leaf free and bridge free clusters for the number of clusters we numerically find the number of leaf free and bridge free clusters on the cg scale as lnv while for 7d they scale as v for the size distribution we find the behavior on the cg is governed by a modified fisher exponent 1 while for leaf free clusters in 7d it is governed by fisher exponent 5 2 the size distribution of bridge free clusters in 7d displays two scaling behavior with exponents 4 and 1 the probability distribution p c 1 v dc 1 of the largest cluster of size c 1 for whole percolation configurations is observed to follow a single variable function p over x dx with x c 1 v d f for both cg and 7d up to a rescaling factor for the variable x the probability functions for cg and 7d collapse on top of each other within the entire range of x the analytical expressions in the x 0 and x limits are further confirmed our work demonstrates that the geometric structure of high dimensional percolation clusters cannot be fully accounted for by their complete graph counterparts

critical random graphs diameter and mixing time let mathcal c 1 denote the largest connected component of the critical erd s renyi random graph g n frac 1 n we show that typically the diameter of mathcal c 1 is of order n1 3 and the mixing time of the lazy simple random walk on mathcal c 1 is of order n the latter answers a question of benjamini kozma and wormald these results extend to clusters of size n2 3 of p bond percolation on any d regular n vertex graph where such clusters exist provided that p d 1 1 o n 1 3

phase transition and critical behavior in a model of organized criticality we study a model of organized criticality where a single avalanche propagates through an a priori static i e organized sandpile configuration the latter is chosen according to an i i d distribution from a borel probability measure on 0 1 the avalanche dynamics is driven by a standard toppling rule however we simplify the geometry by placing the problem on a directed rooted tree as our main result we characterize which are critical in the sense that they do not admit an infinite avalanche but exhibit a power law decay of avalanche sizes our analysis reveals close connections to directed site percolation both in the characterization of criticality and in the values of the critical exponents

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

galton watson trees with the same mean have the same polar sets evans defined a notion of what it means for a set b to be polar for a process indexed by a tree the main result herein is that a tree picked from a galton watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate this implies that deterministic and nondeterministic trees behave identically in a variety of probability models mapping subsets of euclidean space to trees and polar sets to capacity criteria it follows that certain random cantor sets are capacity equivalent to each other and to deterministic cantor sets an extension to branching processes in varying environment is also obtained

random graph asymptotics on high dimensional tori ii volume diameter and mixing time for critical bond percolation on high dimensional torus this paper proves sharp lower bounds on the size of the largest cluster removing a logarithmic correction in the lower bound in heydenreich and van der hofstad comm math phys 270 2 335 358 2007 this improvement finally settles a conjecture by aizenman nuclear phys b 485 3 551 582 1997 about the role of boundary conditions in critical high dimensional percolation and it is a key step in deriving further properties of critical percolation on the torus indeed a criterion of nachmias and peres ann probab 36 4 1267 1286 2008 implies appropriate bounds on diameter and mixing time of the largest clusters we further prove that the volume bounds apply also to any finite number of the largest clusters finally we show that any weak limit of the largest connected component is non degenerate which can be viewed as a significant sign of critical behavior the main conclusion of the paper is that the behavior of critical percolation on the high dimensional torus is the same as for critical erd s renyi random graphs

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

controllability of multiplex multi time scale networks the paradigm of layered networks is used to describe many real world systems from biological networks to social organizations and transportation systems while recently there has been much progress in understanding the general properties of multilayer networks our understanding of how to control such systems remains limited one fundamental aspect that makes this endeavor challenging is that each layer can operate at a different time scale thus we cannot directly apply standard ideas from structural control theory of individual networks here we address the problem of controlling multilayer and multi time scale networks focusing on two layer multiplex networks with one to one interlayer coupling we investigate the practically relevant case when the control signal is applied to the nodes of one layer we develop a theory based on disjoint path covers to determine the minimum number of inputs n i necessary for full control we show that if both layers operate on the same time scale then the network structure of both layers equally affect controllability in the presence of time scale separation controllability is enhanced if the controller interacts with the faster layer n i decreases as the time scale difference increases up to a critical time scale difference above which n i remains constant and is completely determined by the faster layer we show that the critical time scale difference is large if layer i is easy and layer ii is hard to control in isolation in contrast control becomes increasingly difficult if the controller interacts with the layer operating on the slower time scale and increasing time scale separation leads to increased n i again up to a critical value above which n i still depends on the structure of both layers this critical value is largely determined by the longest path in the faster layer that does not involve cycles by identifying the underlying mechanisms that connect time scale difference and controllability for a simplified model we provide crucial insight into disentangling how our ability to control real interacting complex systems is affected by a variety of sources of complexity

random subshifts of finite type let x be an irreducible shift of finite type sft of positive entropy and let bn x be its set of words of length n define a random subset of bn x by independently choosing each word from bn x with some probability let x be the random sft built from the set for each 0 1 and n tending to infinity we compute the limit of the likelihood that x is empty as well as the limiting distribution of entropy for x for near 1 and n tending to infinity we show that the likelihood that x contains a unique irreducible component of positive entropy converges exponentially to 1 these results are obtained by studying certain sequences of random directed graphs this version of random sft differs significantly from a previous notion by the same name which has appeared in the context of random dynamical systems and bundled dynamical systems

random walks capacity and percolation on trees a collection of several different probabilistic processes involving trees is shown to have an unexpected unity this makes possible a fruitful interplay of these probabilistic processes the processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well our work has five specific aims first an exact correspondence between random walks and percolation on trees is proved extending and sharpening previous work of the author this is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes second we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation this capacitary formulation extends and sharpens work of fan on random interval coverings third we show how this formulation also applies to generalize work of evans on random labelling of trees fourth the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a s in particular we resolve a conjecture of griffeath on the necessity of the nash williams criterion fifth for this last purpose we establish several new basic results on branching processes in varying environments

a mathematical construction of the non abelian chern simons functional integral we construct rigorously an infinite dimensional distribution which corresponds to the chern simons cs functional integral associated with a principal fiber bundle over r 3 with structure group a compact connected lie group we determine the moments of the cs distribution and show that these coincide with those used in informal studies of the cs integral a locality property of the cs distribution is proven the complexified theory of frohlich and king is also discussed within our framework

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

chern simons witten invariants of lens spaces and torus bundles and the semiclassical approximation we derive explicit formulas for the chern simons witten invariants of lens spaces and torus bundles overs1 for arbitrary values of the levelk most of our results are for the groupg su 2 though some are for more general compact groups we explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the chern simons path integral

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

conditional independence and chain event graphs graphs provide an excellent framework for interrogating symmetric models of measurement random variables and discovering their implied conditional independence structure however it is not unusual for a model to be specified from a description of how a process unfolds i e via its event tree rather than through relationships between a given set of measurements here we introduce a new mixed graphical structure called the chain event graph that is a function of this event tree and a set of elicited equivalence relationships this graph is more expressive and flexible than either the bayesian network equivalent in the symmetric case or the probability decision graph various separation theorems are proved for the chain event graph these enable implied conditional independencies to be read from the graph s topology we also show how the topology can be exploited to tease out the interesting conditional independence structure of functions of random variables associated with the underlying event tree

random walk on barely supercritical branching random walk let mathcal t be a supercritical galton watson tree with a bounded offspring distribution that has mean mu 1 conditioned to survive let varphi mathcal t be a random embedding of mathcal t into mathbb z d according to a simple random walk step distribution let mathcal t p be percolation on mathcal t with parameter p and let p c mu 1 be the critical percolation parameter we consider a random walk x n n ge 1 on mathcal t p and investigate the behavior of the embedded process varphi mathcal t p x n as n rightarrow infty and simultaneously mathcal t p becomes critical that is p p n searrow p c we show that when we scale time by n p n p c 3 and space by sqrt p n p c n the process varphi mathcal t p x n n ge 1 converges to a d dimensional brownian motion we argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

the chern simons theory and knot polynomials the chern simons gauge theory is studied using a functional integral quantization this leads to a differential equation for expectations of wilson lines the solution of this differential equation is shown to be simply related to the two variable jones polynomial of the corresponding link in the case where the gauge group issu n a similar equation has been used before to get the jones polynomial from a braid representation of the link the main novelty of our approach is that we get the jones polynomial from a plat representation of the link

cluster tails for critical power law inhomogeneous random graphs recently the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank 1 type having finite variance but infinite third moment degrees was obtained in bhamidi et al ann probab 40 2299 2361 2012 it was proved that when the degrees obey a power law with exponent tau in 3 4 the sequence of clusters ordered in decreasing size and multiplied through by n tau 2 tau 1 converges as n rightarrow infty to a sequence of decreasing non degenerate random variables here we study the tails of the limit of the rescaled largest cluster i e the probability that the scaling limit of the largest cluster takes a large value u as a function of u this extends a related result of pittel j combin theory ser b 82 2 237 269 2001 for the erd s renyi random graph to the setting of rank 1 inhomogeneous random graphs with infinite third moment degrees we make use of delicate large deviations and weak convergence arguments

biased random walks on directed trees we define directed rooted labeled and unlabeled trees and find measures on the space of directed rooted unlabeled trees which are invariant with respect to transition probabilities corresponding to a biased random walk on a directed rooted labeled tree we use these to calculate the speed of a biased random walk on directed rooted labeled trees the results are mainly applied to directed trees with recurrent subtrees where the random walker cannot escape

anchored expansion percolation and speed benjamini lyons and schramm random walks and discrete potential theory 1999 56 84 considered properties of an infinite graph g and the simple random walk on it that are preserved by random perturbations in this paper we solve several problems raised by those authors the anchored expansion constant is a variant of the cheeger constant its positivity implies positive lower speed for the simple random walk as shown by virag geom funct anal 10 2000 1588 1605 we prove that if g has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter p sufficiently close to 1 a better estimate for the parameters p where this holds is in the appendix we also show that positivity of the anchored expansion constant is preserved under a random stretch if and only if the stretching law has an exponential tail we then study a simple random walk in the infinite percolation cluster in cayley graphs of certain amenable groups known as lamplighter groups we prove that zero speed for a random walk on a lamplighter group implies zero speed for random walk on an infinite cluster for any supercritical percolation parameter p for p large enough we also establish the converse

monotonicity of uniqueness for percolation on cayley graphs all infinite clusters are born simultaneously consider site or bond percolation with retention parameter p on an infinite cayley graph in response to questions raised by grimmett and newman 1990 and benjamini and schramm 1996 we show that the property of having almost surely a unique infinite open cluster is increasing in p moreover in the standard coupling of the percolation models for all parameters a s for all p2 p1 pc each infinite p2 cluster contains an infinite p1 cluster this yields an extension of alexander s 1995 simultaneous uniqueness theorem as a corollary we obtain that the probability v p that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p pc all our results extend to quasi transitive infinite graphs with a unimodular automorphism group

continuum limit of critical inhomogeneous random graphs the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real world systems in this study we analyze the maximal components at criticality of one famous class of such models the rank one inhomogeneous random graph model norros and reittu adv appl probab 38 1 59 75 2006 bollobas et al random struct algorithms 31 1 3 122 2007 section 16 4 viewing these components as measured random metric spaces under finite moment assumptions for the weight distribution we show that the components in the critical scaling window with distances scaled by n 1 3 converge in the gromov haussdorf prokhorov metric to rescaled versions of the limit objects identified for the erd s renyi random graph components at criticality in addario berry et al probab theory related fields 152 3 4 367 406 2012 a key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called mathbf p trees camarri and pitman electron j probab 5 2 1 18 2000 aldous et al probab theory related fields 129 2 182 218 2004 this is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics braunstein et al phys rev lett 91 16 168701 2003 for such graph models by asymptotic equivalence janson random struct algorithms 36 1 26 45 2010 the same results are true for the chung lu model chung and lu proc natl acad sci 99 25 15879 15882 2002 chung and lu ann combin 6 2 125 145 2002 chung and lu complex graphs and networks 2006 and the britton deijfen martin lof model britton et al j stat phys 124 6 1377 1397 2006 a crucial ingredient of the proof of independent interest are tail bounds for the height of mathbf p trees the techniques developed in this paper form the main technical bedrock for the general program developed in bhamidi et al scaling limits of random graph models at criticality universality and the basin of attraction of the erd s renyi random graph arxiv preprint 2014 for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels bollobas et al random struct algorithms 31 1 3 122 2007

random walks on the random graph we study random walks on the giant component of the erd s renyi random graph g n p g n p where p np n for 1 1 fixed the mixing time from a worst starting point was shown by fountoulakis and reed and independently by benjamini kozma and wormald to have order log2nlog2 n we prove that starting from a uniform vertex equivalently from a fixed vertex conditioned to belong to the giant both accelerates mixing to o logn o log n and concentrates it the cutoff phenomenon occurs the typical mixing is at d 1logn logn 1 2 o 1 d 1log n log n 1 2 o 1 where and dd are the speed of random walk and dimension of harmonic measure on a poisson poisson galton watson tree analogous results are given for graphs with prescribed degree sequences where cutoff is shown both for the simple and for the nonbacktracking random walk

the random cluster model on the complete graph the random cluster model of fortuin and kasteleyn contains as special cases the percolation ising and potts models of statistical physics when the underlying graph is the complete graph onn vertices then the associated processes are called mean field in this study of the mean field random cluster model with parametersp n andq we show that its properties for any value ofq 0 may be derived from those of an erd s renyi random graph in this way we calculate the critical point c q of the model and show that the associated phase transition is continuous if and only ifq 2 exact formulae are given for c q the density of the largest component the density of edges of the model and the free energy this work generalizes earlier results valid for the potts model whereq is an integer satisfyingq 2 equivalent results are obtained for a fixed edge number random cluster model as a consequence of the results of this paper one obtains large deviation theorems for the number of components in the classical random graph models whereq 1

biased random walks on galton watson trees we consider random walks with a bias toward the root on the family tree t of a supercritical galton watson branching process and show that the speed is positive whenever the walk is transient the corresponding harmonic measures are carried by subsets of the boundary of dimension smaller than that of the whole boundary when the bias is directed away from the root and the extinction probability is positive the speed may be zero even though the walk is transient the critical bias for positive speed is determined

knot theory and the heuristics of functional integration abstract this paper is an exposition of the relationship between the heuristics of witten s functional integral and the theory of knots and links in three dimensional space

rotated tcm systems with dual transmit and multiple receive antennas on nakagami fading channels this paper analyzes the bit error rate performance of a dual transmission multiple reception antenna system in conjunction with a rotated trellis coded modulation scheme over nakagami fading channels a simple way to combine the exact calculation of the pairwise error event probability with the transfer function bounding techniques is described this approach allows us to evaluate an extremely tight theoretical upper bound on the average bit error probability the parameters which dominate the performance of these transmission schemes on fading channels are extracted from this evaluation optimum adaptive and fixed phase rotations have been calculated showing that most of the power efficiency obtained by using the optimum adaptive phase rotation can be retained by using the optimum fixed phase rotation

stable processes on the boundary of a regular tree we define a class of processes on the boundary of a regular tree that can be viewed as stable levy processes on z n 0 z n we show that the range of these processes can be compared with a bernoulli percolation as in peres which easily leads to various results on the intersection properties we develop an alternative approach based on the comparison with a branching random walk by this method we establish the existence of points of infinite multiplicity when the index of the process equals the dimension of the state space as for planar brownian motion

martin capacity for markov chains the probability that a transient markov chain or a brownian path will ever visit a given set is classically estimated using the capacity of with respect to the green kernel g x y we show that replacing the green kernel by the martin kernel g x y g 0 y yields improved estimates which are exact up to a factor of 2 these estimates are applied to random walks on lattices and also to explain a connection found by lyons between capacity and percolation on trees

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

functional integration and the theory of knots this paper examines the role of functional integration in the theory of knots and links it is shown how it is possible to find many properties and conjectures about link invariants via this point of view and how these considerations lead to ideas about the mathematical status of such forms of integration

combined coding and modulation theory and applications the theoretical aspects of the encoding process are investigated resulting in a precise definition of linear codes together with theorems that clarify how they can be obtained a particular subset of linear codes called superlinear codes for which the performance analysis is highly simplified is identified the most relevant performance measures for the analysis of this class of codes are discussed the minimum euclidean distance and the event and bit error probabilities are found analytically using the uniform error property when applicable or variations on it this yields accurate upper and lower bounds to the error rate at the price of reasonable computational complexity the theory is then applied to the search for good codes and to their performance evaluation the cases of 16 and 32 psk codes which are good candidates for use in digital satellite transmission are considered several new results in terms of error event and bit error probabilities are presented showing considerable gains in terms of snr with respect to the uncoded case

tree indexed random walks on groups and first passage percolation suppose that i i d random variables are attached to the edges of an infinite tree when the tree is large enough the partial sumss along some of its infinite paths will exhibit behavior atypical for an ordinary random walk this principle has appeared in works on branching random walks first passage percolation and rwre on trees we establish further quantitative versions of this principle which are applicable in these settings in particular different notions of speed for such a tree indexed walk correspond to different dimension notions for trees finally if the labeling variables take values in a group then properties of the group e g polynomial growth or a nontrivial poisson boundary are reflected in the sample path behavior of the resulting tree indexed walk

random walk on sparse random digraphs a finite ergodic markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale originally discovered in the context of card shuffling aldous and diaconis in am math mon 93 333 348 1986 this remarkable phenomenon is now rigorously established for many reversible chains here we consider the non reversible case of random walks on sparse directed graphs for which even the equilibrium measure is far from being understood we work under the configuration model allowing both the in degrees and the out degrees to be freely specified we establish the cutoff phenomenon determine its precise window and prove that the cutoff profile approaches a universal shape we also provide a detailed description of the equilibrium measure

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

random walk on the high dimensional iic we study the asymptotic behavior of the exit times of random walk from euclidean balls around the origin of the incipient infinite cluster in a manner inspired by kumagai and misumi j theor probab 21 910 935 2008 we do this by getting bounds on the effective resistance between the origin and the boundary of these euclidean balls we show that the geometric properties of long range percolation clusters are significantly different from those of finite range clusters we also study the behavior of random walk on the backbone of the iic and we prove that the alexander orbach conjecture holds for the incipient infinite cluster in high dimensions both for long range percolation and for finite range percolation

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

vassiliev knot invariants and chern simons perturbation theory to all orders at any order the perturbative expansion of the expectation values of wilson lines in chern simons theory gives certain integral expressions we show that they all lead to knot invariants moreover these are finite type invariants whose order coincides with the order in the perturbative expansion together they combine to give a universal vassiliev invariant

cover times blanket times and majorizing measures we exhibit a strong connection between cover times of graphs gaussian processes and talagrand s theory of majorizing measures in particular we show that the cover time of any graph g is equivalent up to universal constants to the square of the expected maximum of the gaussian free field on g scaled by the number of edges in g this allows us to resolve a number of open questions we give a deterministic polynomial time algorithm that computes the cover time to within an o 1 factor for any graph answering a question of aldous and fill 1994 we also positively resolve the blanket time conjectures of winkler and zuckerman 1996 showing that for any graph the blanket and cover times are within an o 1 factor the best previous approximation factor for both these problems was o loglogn 2 for n vertex graphs due to kahn kim lovasz and vu 2000

constellation design for trellis coded unitary space time modulation systems we consider the problem of creating signal constellations for trellis coded unitary space time communication links where neither the transmitter nor the receiver knows the fading gains of the channel our study includes constellation design techniques for trellis coded schemes with and without parallel paths which allows us to find a tradeoff between low complexity and high performance we present a new formulation of the constellation design problem for trellis coded unitary space time modulation tcustm schemes the two key differences in our approach against those of other authors are that we not only combine the constellation design and mapping by set partitioning into one step but we also use directly the chernoff bound of the pairwise error probability as a design metric by novelly employing a theorem for the clarke subdifferential of the sum of the k largest singular values of the unitary matrix we also present a numerical optimization procedure for finding signal constellations resulting in high performance communications systems to demonstrate the advantages of our new design method we report the best constellations found for tcustm systems simulation results show that these constellations achieve a 1 db coding gain at a bit error rate of 10 4 against usually used constellations

perturbative expansion of chern simons theory with non compact gauge group naive imitation of the usual formulas for compact gauge group in quantizing three dimensional chern simons gauge theory with non compact gauge group leads to formulas that are wrong or unilluminating in this paper an appropriate modification is described which puts the perturbative expansion in a standard manifestly unitary format the one loop contributions which differ from naive extrapolation from the case of compact gauge group are computed and their topological invariance is verified

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

dynamical percolation on general trees haggstrom et al ann inst h poincare probab stat 33 4 497 528 1997 have introduced a dynamical version of percolation on a graph g when g is a tree they derived a necessary and sufficient condition for percolation to exist at some time t in the case that g is a spherically symmetric tree peres and steif in probab theory relat fields 111 1 141 165 1998 derived a necessary and sufficient condition for percolation to exist at some time t in a given target set d the main result of the present paper is a necessary and sufficient condition for the existence of percolation at some time t in d in the case that the underlying tree is not necessary spherically symmetric this answers a question of yuval peres personal communication we present also a formula for the hausdorff dimension of the set of exceptional times of percolation

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

almost sure convergence for stochastically biased random walks on trees we are interested in the biased random walk on a supercritical galton watson tree in the sense of lyons ann probab 18 931 958 1990 and lyons pemantle and peres probab theory relat fields 106 249 264 1996 and study a phenomenon of slow movement in order to observe such a slow movement the bias needs to be random the resulting random walk is then a tree valued random walk in random environment we investigate the recurrent case and prove under suitable general integrability assumptions that upon the system s non extinction the maximal displacement of the walk in the first n steps divided by log n 3 converges almost surely to a known positive constant

mixing time of near critical random graphs let mathcal c 1 be the largest component of the erd s renyi random graph mathcal g n p the mixing time of random walk on mathcal c 1 in the strictly supercritical regime p c n with fixed c 1 was shown to have order log2n by fountoulakis and reed and independently by benjamini kozma and wormald in the critical window p 1 e n where e3n is bounded nachmias and peres proved that the mixing time on mathcal c 1 is of order n however it was unclear how to interpolate between these results and estimate the mixing time as the giant component emerges from the critical window indeed even the asymptotics of the diameter of mathcal c 1 in this regime were only recently obtained by riordan and wormald as well as the present authors and kim in this paper we show that for p 1 e n with e3n and o n the mixing time on mathcal c 1 is with high probability of order n log2 in addition we show that this is the order of the largest mixing time over all components both in the slightly supercritical and in the slightly subcritical regime i e p 1 e n with as above

domination between trees and application to an explosion problem we define a notion of stochastic domination between trees where one tree dominates another if when the vertices of each are labeled with independent identically distributed random variables one tree is always more likely to contain a path with a specified property sufficient conditions for this kind of domination are 1 more symmetry and 2 earlier branching we apply these conditions to the problem of determining how fast a tree must grow before first passage percolation on the tree exhibits an explosion that is to say infinitely many vertices are reached in finite time for a tree in which each vertex at distance n 1 from the root has f n offspring f nondecreasing an explosion occurs with exponentially distributed passage times if and only if f n 1

conceptual proofs of l log l criteria for mean behavior of branching processes the kesten stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process showing that an l log l condition is decisive in critical and subcritical cases results of kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations we give conceptual proofs of these theorems based on comparisons of galton watson measure to another measure on the space of trees this approach also explains yaglom s exponential limit law for conditioned critical branching processes via a simple characterization of the exponential distribution

kontsevich integral for vassiliev invariants from chern simons perturbation theory in the light cone gauge we analyze the structure of the perturbative series expansion of chern simons gauge theory in the light cone gauge after introducing a regularization prescription that entails the consideration of framed knots we present the general form of the vacuum expectation value of a wilson loop the resulting expression turns out to give the same framing dependence as the one obtained using nonperturbative methods and perturbative methods in covariant gauges it also contains the kontsevich integral for vassiliev invariants of framed knots

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods

kakeya sets and directional maximal operators in the plane we completely characterize the boundedness of planar directional maximal operators on lp more precisely if is a set of directions we show that m the maximal operator associated to line segments in the directions is unbounded on lp for all p 1

on the self linking of knots this note describes a subcomplex f of the de rham complex of parametrized knot space which is combinatorial over a number of universal anomaly integrals the self linking integrals of guadaguini martellini and mintchev perturbative aspects of chern simons field theory phys lett b 227 111 1989 and bar natan perturbative aspects of the chern simons topological quantum field theory ph d thesis princeton university 1991 also on the vassiliev knot invariants to appear in topology are seen to represent the first nontrivial element in h0 f occurring at level 4 and are anomaly free however already at the next level an anomalous term is possible

biased random walks on galton watson trees with leaves we consider a biased random walk xn on a galton watson tree with leaves in the sub ballistic regime we prove that there exists an explicit constant 0 1 depending on the bias such that xn is of order n denoting n the hitting time of level n we prove that n n1 is tight moreover we show that n n1 does not converge in law at least for large values of we prove that along the sequences n k k n n1 converges to certain infinitely divisible laws key tools for the proof are the classical harris decomposition for galton watson trees a new variant of regeneration times and the careful analysis of triangular arrays of i i d heavy tailed random variables

the critical ising model on trees concave recursions and nonlinear capacity we consider the ising model on a general tree under various boundary conditions all plus free and spin glass in each case we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity we obtain exact capacity criteria that govern behavior at critical temperatures for plus boundary conditions an l 3 capacity arises in particular on a spherically symmetric tree that has n b n vertices at level n up to bounded factors we prove that there is a unique gibbs measure for the ferromagnetic ising model at the relevant critical temperature if and only if 1 2 our proofs are based on a new link between nonlinear recursions on trees and l p capacities

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free