## Shape fluctuations are different in different directions.(English)Zbl 1130.60093

Summary: We consider the first passage percolation model on $$\mathbb Z^2$$. In this model, we assign independently to each edge $$e$$ a passage time $$t(e)$$ with a common distribution $$F$$. Let $$T(u, v)$$ be the passage time from $$u$$ to $$v$$. In this paper, we show that, whenever $$F(0)<p_c$$, $$\sigma^2(T((0, 0),(n,0)))\geq C\log n$$ for all $$n\geq 1$$. Note that if $$F$$ satisfies an additional special condition, $$\inf\text{supp}(F)=r>0$$ and $$F(r)>\vec p_c$$, it is known that there exists $$M$$ such that for all $$n$$, $$\sigma_2(T((0, 0), (n, n)))\leq M$$. These results tell us that shape fluctuations not only depend on the distribution $$F$$, but also on direction. When showing this result, we find the following interesting geometrical property. With the special distribution above, any long piece with $$r$$-edges in an optimal path from $$(0, 0)$$ to $$(n, 0)$$ has to be very circuitous.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory

### Keywords:

first passage percolation; fluctuations
Full Text:

### References:

 [1] Alexander, K. (1993). A note on some rates of convergence in first-passage percolation. Ann. Appl. Probab. 3 81–91. · Zbl 0771.60090 · doi:10.1214/aoap/1177005508 [2] Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab. 31 1970–1978. · Zbl 1087.60070 · doi:10.1214/aop/1068646373 [3] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12 999–1040. · Zbl 0567.60095 · doi:10.1214/aop/1176993140 [4] Durrett, R. and Liggett, T. M. (1981). The shape of the limit set in Richardson’s growth model. Ann. Probab. 9 186–193. · Zbl 0457.60083 · doi:10.1214/aop/1176994460 [5] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli , Bayes , Laplace Anniversary Volume (J. Neyman and L. LeCam, eds.) 61–110. Springer, Berlin. · Zbl 0143.40402 [6] Kesten, H. (1986). Aspects of first-passage percolation. École d’Été de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180 125–264. Springer, Berlin. · Zbl 0602.60098 · doi:10.1007/BFb0074919 [7] Kesten, H. (1993). On the speed of convergence in first passage percolation. Ann. Appl. Probab. 3 296–338. · Zbl 0783.60103 · doi:10.1214/aoap/1177005426 [8] Kesten, H. and Zhang, Y. (1990). The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 537–555. · Zbl 0705.60092 · doi:10.1214/aop/1176990844 [9] Kesten, H. and Zhang, Y. (1997). A central limit theorem for critical first passage percolation in two dimensions. Probab. Theory Related Fields 107 137–160. · Zbl 0868.60077 · doi:10.1007/s004400050080 [10] Krug, J. and Spohn, H. (1992). Kinetic roughening of growing surfaces. In Solids Far from Equilibrium : Growth , Morphology , Defects (C. Godreche, ed.) 497–582. Cambridge Univ. Press. [11] Marchand, R. (2002). Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 1001–1038. · Zbl 1062.60100 · doi:10.1214/aoap/1031863179 [12] Newman, C. and Piza, M. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977–1005. · Zbl 0835.60087 · doi:10.1214/aop/1176988171 [13] Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice . Lecture Notes in Math. 671 . Springer, Berlin. · Zbl 0379.60001 · doi:10.1007/BFb0063306 [14] Yukich, J. and Zhang, Y. (2006). Singularity points for first passage percolation. Ann. Probab. 34 577–592. · Zbl 1097.60084 · doi:10.1214/009117905000000819 [15] Zhang, Y. (1999). Double behavior of critical first-passage percolation. In Perplexing Problems in Probability (M. Bramson and R. Durrett, eds.) 143–158. Birkhäuser, Boston. · Zbl 0941.60094 [16] Zhang, Y. (2005). On the speeds of convergence and concentration of a subadditive ergodic process. [17] Zhang, Y. (2006). The divergence of fluctuations for shape in first passage percolation. Probab. Theory Related Fields 136 298–320. · Zbl 1097.60085 · doi:10.1007/s00440-005-0488-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.