jamming at zero temperature and zero applied stress the epitome of disorder we have studied how two and three dimensional systems made up of particles interacting with finite range repulsive potentials jam i e develop a yield stress in a disordered state at zero temperature and zero applied stress at low packing fractions phi the system is not jammed and each particle can move without impediment from its neighbors for each configuration there is a unique jamming threshold phi c at which particles can no longer avoid each other and the bulk and shear moduli simultaneously become nonzero the distribution of phi c values becomes narrower as the system size increases so that essentially all configurations jam at the same packing fraction in the thermodynamic limit this packing fraction corresponds to the previously measured value for random close packing in fact our results provide a well defined meaning for random close packing in terms of the fraction of all phase space with inherent structures that jam the jamming threshold point j occurring at zero temperature and applied stress and at the random close packing density has properties reminiscent of an ordinary critical point as point j is approached from higher packing fractions power law scaling is found for the divergence of the first peak in the pair correlation function and in the vanishing of the pressure shear modulus and excess number of overlapping neighbors moreover near point j certain quantities no longer self average suggesting the existence of a length scale that diverges at j however point j also differs from an ordinary critical point the scaling exponents do not depend on dimension but do depend on the interparticle potential finally as point j is approached from high packing fractions the density of vibrational states develops a large excess of low frequency modes indeed at point j the density of states is a constant all the way down to zero frequency all of these results suggest that point j is a point of maximal disorder and may control behavior in its vicinity perhaps even at the glass transition

jamming graphs a local approach to global mechanical rigidity we revisit the concept of minimal rigidity as applied to frictionless repulsive soft sphere packings in two dimensions with the introduction of the jamming graph minimal rigidity is a purely combinatorial property encoded via laman s theorem in two dimensions it constrains the global average coordination number of the graph for example however minimal rigidity does not address the geometry of local mechanical stability the jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability we demonstrate how jamming graphs can be constructed using local moves via the henneberg construction such that these graphs fall under the jurisdiction of correlated percolation we then probe how jamming graphs destabilize or become unjammed by deleting a bond and computing the resulting rigid cluster distribution we also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra redundant contacts destabilizes the latter endeavor allows us to probe a disk packing in the rigid phase and uncover a potentially new diverging length scale associated with the random deletion of contacts as compared to the study of cut out or frozen in subsystems

culling avalanches in bootstrap percolation we study the culling avalanches which occur after the death of a single randomly chosen site in a network where sites are unstable and are culled if they have coordination less than an integer parameter m avalanche distributions are presented for triangular and cubic lattices for values of m where the associated bootstrap transitions are either first or second order in second order cases the culling avalanche distribution is found to be exponential while in first order cases it follows a power law we present an exact relation between culling avalanches and conventional bootstrap percolation and show that a relation proposed by manna physica a 261 351 1998 can be a good approximation for strongly first order bootstrap transitions but not for continuous bootstrap transitions

k core percolation in four dimensions the k core percolation on the bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first order second order nature we investigate numerically k core percolation on the four dimensional regular lattice for k 4 the presence of a discontinuous transition is clearly established but its nature is strictly first order in particular the k core density displays no singular behavior before the jump and its correlation length remains finite for k 3 the transition is continuous

fast monte carlo algorithm for site or bond percolation we describe in detail an efficient algorithm for studying site or bond percolation on any lattice the algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system we demonstrate our algorithm by using it to investigate a number of issues in percolation theory including the position of the percolation transition for site percolation on the square lattice the stretched exponential behavior of spanning probabilities away from the critical point and the size of the giant component for site percolation on random graphs

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

a not so simple local multiplier algebra we construct an af algebra a such that its local multiplier algebra mloc a does not agree with mloc mloc a thus answering a question raised by g k pedersen in 1978

the local multiplier algebra of a c algebra ii abstract let a be a separable c algebra and let m loc a be the local multiplier algebra of a it is shown that every minimal closed prime ideal of m loc a is primitive if prim a has a dense g consisting of closed points for instance if prim a is a t 1 space and a is unital then m loc a is its own local multiplier algebra and has only inner derivations the same is true for m loc m loc a if a is non unital if a is postliminal then m loc m loc a is the regular completion of a which is an aw algebra

second order local multiplier algebras of continuous trace c algebras abstract we determine the injective envelope and local multiplier algebra of a continuous trace c algebra a that arises from a continuous hilbert bundle over an arbitrary locally compact hausdorff space in addition we show that the second order local multiplier algebra m loc 2 a of any such algebra a is injective

evidence for non universal exponents in bootstrap percolation bootstrap percolation bp models are systems where sites are initially randomly occupied those sites that do not maintain a suitable local environment of at least m occupied sites are successively removed current open questions concerning these systems now focus on the universality of critical exponents for the lower m values simulations indicate that for m 3 on the simple cubic lattice the exponent beta which characterizes the critical behaviour of the percolation probability is distinct from that of usual random percolation m 0 the authors quote pc 0 5717 or 0 0005 and beta 0 6 or 0 1 for m 3 they argue that the connectivity exponent nu for m 2 should be the same as that of usual random percolation and within the limits of numerical accuracy the authors observe that this appears to be the case for m 3 on the simple cubic lattice the implications of this surprising dichotomy are considered

the local multiplier algebra of a c algebra with finite dimensional irreducible representations abstract we describe up to isomorphism the local multiplier algebra m loc a of a c algebra a which admits only finite dimensional irreducible representations m loc a in this case is a finite or countable direct product of c algebras of the form c x n m n where each space x n is stonean in particular m loc a is an a w algebra of type i so it admits only inner derivations and it coincides with the injective envelope i a of a

on the study of jamming percolation we investigate kinetically constrained models of glassy transitions and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales the models we investigate have constraints similar to that of the knights model introduced by toninelli biroli and fisher tbf but differing neighbor relations we find that such knights like models otherwise known as models of jamming percolation need a no parallel crossing rule for the tbf proof of a glassy transition to be valid furthermore most knights like models fail a no perpendicular crossing requirement and thus need modification to be made rigorous we also show how the no parallel crossing requirement can be used to evaluate the provable glassiness of other correlated percolation models by looking at models with more stable directions than the knights model finally we show that the tbf proof does not generalize in any straightforward fashion for three dimensional versions of the knights like models

force balance percolation we study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance i e for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions we prove rigorously that p c 1 for the two dimensional models studied numerical data indicate that the force balance percolation transition is discontinuous with a growing crossover length with perhaps the same form as the jamming percolation models suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases we find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition finally we study the dynamics of the culling procedure invoked to obtain the force balance configurations and find a dynamical exponent similar to that found in sandpile models

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

proof of straley s argument for bootstrap percolation we prove thatpc 1 for bootstrap percolation with large void instabilities in particular ifm 3 on the square lattice

correlated percolation and tricriticality the recent proliferation of correlated percolation models models where the addition of edges and or vertices is no longer independent of other edges and or vertices has been motivated by the quest to find discontinuous percolation transitions the leader in this proliferation is what is known as explosive percolation a recent proof demonstrates that a large class of explosive percolation type models does not in fact exhibit a discontinuous transition riordan and warnke science 333 322 2011 here we discuss two lesser known correlated percolation models the k 3 core model on random graphs and the counter balance model in two dimensions both exhibiting discontinuous transitions to search for tricriticality we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition using a powerful rate equation approach we demonstrate that a mixture of k 2 core and k 3 core vertices on the random graph exhibits a tricritical point however for a mixture of k core and counter balance vertices in two dimensions as the fraction of counter balance vertices is increased numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition i e when all vertices are counter balanced interestingly these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions in addition our results may have potential implications for glassy and jamming systems

universality thresholds and critical exponents in correlated percolation abstract correlated percolation models are systems where sites in a lattice are occupied randomly by a given species and then species are removed bootstrap percolation or added diffusion percolation according to the site s environment results for critical concentrations and exponents of bootstrap and diffusion site percolation models are presented for the square and honeycomb lattices calculations are based on numerical simulation results and are consistent with universal exponents for random and correlated percolation in these lattices

points of differentiability of the norm in lipschitz free spaces abstract we consider convex series of molecules in lipschitz free spaces i e elements of the form n n x n y n d x n y n such that n n we characterise these elements in terms of geometric conditions on the points x n y n of the underlying metric space and determine when they are points of g teaux differentiability of the norm in particular we show that g teaux and frechet differentiability are equivalent for finitely supported elements of lipschitz free spaces over uniformly discrete and bounded metric spaces and that their tensor products with g teaux resp frechet differentiable elements of a banach space are g teaux resp frechet differentiable in the corresponding projective tensor product

universality in bootstrap and diffusion percolation critical concentrations and exponents of bootstrap and diffusion site percolation models are presented for the triangular lattice results are based on numerical simulations and are consistent with universal exponents for random bootstrap and diffusion percolation