scholarly journals Frog models on trees through renewal theory

2018 ◽  
Vol 55 (3) ◽  
pp. 887-899 ◽  
Author(s):  
Sandro Gallo ◽  
Pablo M. Rodriguez
Keyword(s):  
Random Walks ◽  
Critical Parameter ◽  
Renewal Theory ◽  
Percolation Model ◽  
Critical Parameters

Abstract We study a class of growing systems of random walks on regular trees, known as frog models with geometric lifetime in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improves the existing bounds for the critical parameter of a percolation model on trees known as cone percolation.

scholarly journals Approximating Critical Parameters of Branching Random Walks

2009 ◽  
Vol 46 (02) ◽  
pp. 463-478 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Critical Parameter ◽  
Branching Random Walk ◽  
Critical Parameters ◽  
Bond Percolation ◽  
Branching Random Walks

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

scholarly journals Approximating Critical Parameters of Branching Random Walks

2009 ◽  
Vol 46 (2) ◽  
pp. 463-478 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Critical Parameter ◽  
Branching Random Walk ◽  
Critical Parameters ◽  
Bond Percolation ◽  
Branching Random Walks

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

2001 ◽  
Vol 33 (3) ◽  
pp. 652-673 ◽  
Author(s):  
Cheng-Der Fuh ◽  
Tze Leung Lai
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Renewal Theory ◽  
Passage Time ◽  
Boundary Crossing ◽  
High Threshold ◽  
Renewal Theorem ◽  
First Passage ◽  
Markov Random ◽  
General State Space

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk (Xn,Sn), n ≥0, in which Xn takes values in a general state space and Sn takes values in ℝd. In particular, for the case d = 1, we use this result to derive an asymptotic formula for the variance of the first passage time when Sn exceeds a high threshold b, generalizing Smith's classical formula in the case of i.i.d. positive increments for Sn. For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (XT,ST), where T = inf { n : Sn,1 > b } and Sn,1 denotes the first component of Sn.

scholarly journals New Aspects on the Direct Solid State Polycondensation (DSSP) of Aliphatic Nylon Salts: The Case of Hexamethylene Diammonium Dodecanoate

Polymers ◽  
2021 ◽  
Vol 13 (16) ◽  
pp. 2625
Author(s):  
Angeliki D. Mytara ◽  
Athanasios D. Porfyris ◽  
Stamatina N. Vouyiouka ◽  
Constantine D. Papaspyrides
Keyword(s):  
Molecular Weight ◽  
Solid State ◽  
Critical Parameter ◽  
Critical Parameters ◽  
Reaction Conditions ◽  
Solid State Polymerization ◽  
Salt Crystals ◽  
Direct Solid ◽  
End Group ◽  
Aliphatic Polyamide

The direct solid state polymerization (DSSP) of hexamethylene diammonium dodecanoate (PA 612 salt) was investigated for two different salt grades, fossil-based and bio-based. Aliphatic polyamide salts (such as PA 612 salt) are highly susceptible to solid melt transition (SMT) phenomena, which restrain the industrial application of DSSP. To that end, emphasis was given on reactor design, being the critical parameter influencing byproduct diffusion, amine loss and inherent DSSP kinetics. Experiments took place both at the microscale and the laboratory scale, in which two different reactors were tested in terms of bypassing SMT phenomena. The new reactor designed here proved quite successful in maintaining the solid state during the reaction. Scouting experiments were conducted in order to assess the effect of critical parameters and determine appropriate reaction conditions. Fossil-based PA 612 products proved to have a better end-group imbalance in comparison to bio-based ones, which is critical in terms of achieving high molecular weight. Finally, a real DSSP process was demonstrated, starting from PA 612 salt crystals and ending with PA 612 particles.

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