scholarly journals Random Walks Reaching Against all Odds the other Side of the Quarter Plane

2013 ◽  
Vol 50 (1) ◽  
pp. 85-102 ◽  
Author(s):  
Johan S. H. van Leeuwaarden ◽  
Kilian Raschel

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state (i0,j0), we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive a certain integral representation for the probability of this event, and an asymptotic expression for the case when i0 becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model, and the asymmetric exclusion process.

2013 ◽  
Vol 50 (01) ◽  
pp. 85-102 ◽  
Author(s):  
Johan S. H. van Leeuwaarden ◽  
Kilian Raschel

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state (i0,j0), we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive a certain integral representation for the probability of this event, and an asymptotic expression for the case wheni0becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model, and the asymmetric exclusion process.


2017 ◽  
Author(s):  
Khanh Dao Duc ◽  
Zain H. Saleem ◽  
Yun S. Song

AbstractThe Totally Asymmetric Exclusion Process (TASEP) is a classical stochastic model for describing the transport of interacting particles, such as ribosomes moving along the mRNA during translation. Although this model has been widely studied in the past, the extent of collision between particles and the average distance between a particle to its nearest neighbor have not been quantified explicitly. We provide here a theoretical analysis of such quantities via the distribution of isolated particles. In the classical form of the model in which each particle occupies only a single site, we obtain an exact analytic solution using the Matrix Ansatz. We then employ a refined mean field approach to extend the analysis to a generalized TASEP with particles of an arbitrary size. Our theoretical study has direct applications in mRNA translation and the interpretation of experimental ribosome profiling data. In particular, our analysis of data from S. cerevisiae suggests a potential bias against the detection of nearby ribosomes with gap distance less than ~ 3 codons, which leads to some ambiguity in estimating the initiation rate and protein production flux for a substantial fraction of genes. Despite such ambiguity, however, we demonstrate theoretically that the interference rate associated with collisions can be robustly estimated, and show that approximately 1% of the translating ribosomes get obstructed.


2010 ◽  
Vol 47 (02) ◽  
pp. 562-571
Author(s):  
Elcio Lebensztayn ◽  
Fábio Prates Machado ◽  
Mauricio Zuluaga Martinez

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.


2010 ◽  
Vol 47 (2) ◽  
pp. 562-571 ◽  
Author(s):  
Elcio Lebensztayn ◽  
Fábio Prates Machado ◽  
Mauricio Zuluaga Martinez

We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.


2005 ◽  
Vol 19 (18) ◽  
pp. 2989-2998 ◽  
Author(s):  
RUI JIANG ◽  
YAO-MING YUAN ◽  
QING-SONG WU

In this paper, we have investigated the interplay of the asymmetric exclusion process (ASEP) of parallel updated rules with "Langmuir kinetics" (LK) by equipping the ASEP system with the additional feature of local particle creation at empty sites with rate ωa and annihilation with rate ωd. Two slightly different models are proposed. It is shown that the phase separation may occur in both models. However, in one model, the maximum flow phase may occur but in the other model, there only exists the constant bulk density phase.


1968 ◽  
Vol 25 (2) ◽  
pp. 393-407 ◽  
Author(s):  
John R. Hunter

Schools of six jack mackerel each were photographed with infrared film at eight levels of luminance and also in darkness. Three indices were used to measure the behavior of the school from motion pictures. Two of the indices, mean distance to nearest neighbor and mean separation distance, were measures of the distances between individuals in a school; the other, mean angular deviation, was a measure of differences in orientation between individuals. A value for each index was calculated for each motion picture frame.From 12.1 to 6 × 10−6 ft-L no differences existed in the angular deviation of the school or in the distances between fish. At 6 × 10−7 ft-L the intervals between fish were much larger than at higher levels of brightness and groups showed little uniformity in their orientation. Below 6 × 10−7 ft-L (darkness) schools were dispersed and the distributions of values of angular deviation were random.The ability of jack mackerel to feed on live adult Artemia was also tested at eight levels of luminance and in darkness. The number of Artemia eaten at 6 × 10−5 ft-L was about half of that eaten at the normal daytime level of 12.1 ft-L. Few Artemia were eaten at 6 × 10−7 ft-L and none in darkness.Comparison of these data with measurements of light in the sea indicated that jack mackerel probably would be able to maintain schools near the surface on a moonless starlit night and that they probably could feed effectively near the surface on a full moonlight night.


1993 ◽  
Vol 07 (29n30) ◽  
pp. 1947-1950 ◽  
Author(s):  
RAFFAELLA BURIONI ◽  
DAVIDE CASSI

We rigorously prove that the correlation functions of any statistical model having a compact transitive symmetry group and nearest-neighbor interactions on any tree structure are equal to the corresponding ones on a linear chain. The exponential decay of the latter implies the absence of long-range order on any tree. On the other hand, for trees with exponential growth such as Bethe lattices, one can show the existence of a particular kind of mean field phase transition without long-range order.


1940 ◽  
Vol 30 (2) ◽  
pp. 139-178
Author(s):  
J. Emilio Ramirez

Summary Over a period of six months, from July to December, 1938, an investigation on microseismic waves has been carried out in the Department of Geophysics of St. Louis University. Four electromagnetic seismographs, specially designed for recording microseisms, were installed in the city of St. Louis in the form of a triangular network. Two of these were E-W components, one at the St. Louis University Gymnasium and the other 6.4 km. due west at Washington University. The other two were arranged as N-S components, one at the St. Louis University Gymnasium and one 6.3 km. due south at Maryville College. The speed of the photographic paper was 60 mm/min., and time signals were recorded automatically and simultaneously on each paper from the same clock every minute and at shorter intervals from a special pendulum and “tickler” combination by means of telephone wires. The results have demonstrated beyond doubt that microseismic waves are traveling and not stationary waves. The same waves have been identified at each one of the stations of the network, and also at Florissant, 21.8 km. away from St. Louis University. The speed of microseismic waves at St. Louis was determined from several storms of microseisms and it was found to be 2.67±0.03 km/sec. The direction of microseisms was also established for most of the storms and it was found that about 80 per cent of incoming microseisms at St. Louis were from the northeast quadrant during the interval from July to December, 1938. No microseisms were recorded from the south, west, or southwest. The period of the waves varied between 3.5 and 7.5 sec. The average period was about 5.4 sec. The microseismic wave length was therefore of the order of 14¼ km. A study of the nature of microseismic waves from the three Galitzin-Wilip components of the Florissant station reveals in the waves many of the characteristics of the Rayleigh waves; that is, the particles in the passage of microseismic waves move in elliptical orbits of somewhat larger vertical axis and with retrograde motion. A comparison carried over a period of more than a year between microseisms and microbarometric oscillations recorded by specially designed microbarographs showed no direct relationship between the two phenomena in wave form, group form, period, or duration of storms. The source of microseisms is to be found not over the land, but rather out over the surface of the ocean. The amplitudes of microseisms depend only on the intensity and widespread character of barometric lows traveling over the ocean. Several correlations between the two phenomena seem to make this conclusion rather evident. Special emphasis is laid on the fact that all the determined directions of incoming microseisms at St. Louis point to a deep barometric low over the ocean. The period of microseisms seems to be a function of the distance between the station and the source of microseisms. The exact mechanism by which barometric lows over the ocean water result in the production of microseisms needs further investigation. Large microseisms have been produced without any indication of surf near the coasts, or with winds blowing from the land toward the ocean.


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