We outline the properties of a symmetric random walk in one dimension in which the length of the step equals with As the number of steps the probability that the end point is at approaches a limiting distribution that has many beautiful features. For the support of is a Cantor set. For there is a countably infinite set of λ values for which is singular, while is smooth for almost all other λ values. In the most interesting case of is riddled with singularities and is strikingly self-similar. This self-similarity is exploited to derive a simple form for the probability measure
REFERENCES
1.
B.
Jessen
and A.
Wintner
, “Distribution functions and the Riemann zeta function
,” Trans. Am. Math. Soc.
38
, 48
–88
(1935
);B.
Kershner
and A.
Wintner
, “On symmetric Bernoulli convolutions
,” Am. J. Math.
57
, 541
–548
(1935
);A.
Wintner
, “On convergent Poisson convolutions
,” Am. J. Math.
57
, 827
–838
(1935
).2.
P.
Erdös
, “On a family of symmetric Bernoulli convolutions
,” Am. J. Math.
61
, 974
–976
(1939
);P.
Erdös
, “On smoothness properties of a family of Bernoulli convolutions
,” Am. J. Math.
62
, 180
–186
(1940
).3.
A. M.
Garsia
, “Arithmetic properties of Bernoulli convolutions
,” Trans. Am. Math. Soc.
102
, 409
–432
(1962
);A. M.
Garsia
, “Entropy and singularity of infinite convolutions
,” Pac. J. Math.
13
, 1159
–1169
(1963
).4.
M. Kac, Statistical Independence in Probability, Analysis and Number Theory (Mathematical Association of America; distributed by Wiley, New York, 1959).
5.
E.
Barkai
and R.
Silbey
, “Distribution of single-molecule line widths
,” Chem. Phys. Lett.
310
, 287
–295
(1999
).6.
E.
Ben-Naim
, S.
Redner
, and D.
ben-Avraham
, “Bimodal diffusion in power-law shear flows
,” Phys. Rev. A
45
, 7207
–7213
(1992
).7.
J. C.
Alexander
and J. A.
Yorke
, “Fat baker’s transformations
,” Ergod. Theory Dyn. Syst.
4
, 1
–23
(1984
);J. C.
Alexander
and D.
Zagier
, “The entropy of a certain infinitely convolved Bernoulli measure
,” J. Lond. Math. Soc.
44
, 121
–134
(1991
).8.
F.
Ledrappier
, “On the dimension of some graphs
,” Contemp. Math.
135
, 285
–293
(1992
).9.
Y. Peres, W. Schlag, and B. Solomyak, “Sixty years of Bernoulli convolutions,” in Fractals and Stochastics II, edited by C. Bandt, S. Graf, and M. Zähle, Progress in Probability (Birkhauser, Boston, 2000), Vol. 46, pp. 39–65.
10.
P.
Diaconis
and D.
Freedman
, “Iterated random functions
,” SIAM Rev.
41
, 45
–76
(1999
).11.
A. C.
de la Torre
, A.
Maltz
, H. O.
Mártin
, P.
Catuogno
, and I.
Garciá-Mata
, “Random walk with an exponentially varying step
,” Phys. Rev. E
62
, 7748
–7754
(2000
).12.
F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw–Hill, New York, 1965).
13.
G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam 1994); S. Redner, A Guide to First-Passage Processes (Cambridge U.P., New York, 2001).
14.
See, for example, lecture notes by M. Bazant for MIT course 18.366. The URL is 〈http://www-math.mit.edu/18.366/〉.
15.
B.
Solomyak
, “On the random series (an Erdös problem)
,” Ann. Math.
142
, 611
–625
(1995
).16.
More details about Pisot numbers can be found in M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, and M. Pathiaux-Delefosse, Pisot and Salem Numbers (Birkhäuser, Boston, 1992).
17.
F.
Ledrappier
and A.
Porzio
, “A dimension formula for Bernoulli convolutions
,” J. Stat. Phys.
76
, 1307
–1327
(1994
).18.
N.
Sidorov
and A.
Vershik
, “Ergodic properties of Erds measure, the entropy of the goldenshift, and related problems
,” Monatsh. Math.
126
, 215
–261
(1998
).19.
G. B. Arfken, Mathematical Methods for Physicists (Academic, San Diego, 1995).
20.
See, for example, N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
21.
J. H. Conway and R. K. Guy, The Book of Numbers (Springer, New York, 1996).
22.
Y.
Peres
and B.
Solomyak
, “Self-similar measures and intersections of Cantor sets
,” Trans. Am. Math. Soc.
350
, 4065
–4087
(1998
).23.
M. Bazant, B. Bradley, and J. Choi (unpublished).
24.
B. Bradley (private communication).
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