Timeline for probabilities of increasing events under different product measures.
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Jun 2, 2010 at 22:27 | comment | added | Ryan O'Donnell | The rough philosophy of Friedgut's results is that events A which do not "essentially depend on a small number of coordinates" have the property that the derivative of $P_x(A)$ is large at the "critical probability" (i.e., the $x_c$ such that $P_{x_c}(A)$ is $1/2$). Thus the event has probability nearly $0$ or $1$ for almost all $x$. | |
| Jun 2, 2010 at 11:21 | comment | added | James Martin | My impression is that the power of Friedgut's results is when the number of variables becomes large. Perhaps here, in looking for a bound that is uniform in $A$, it will be events that depend (or nearly depend) on a small number of variables that are the most important. | |
| Jun 2, 2010 at 11:19 | comment | added | James Martin | Ryan, the question now becomes: can one find a bound on $\sum_i P_x(A_i)$ in terms of $P_x(A)$? In particular, if $P_x(A)$ is not too close to 0 or 1, does it follow that $\sum_i P_x(A_i)$ is reasonably large? | |
| Jun 2, 2010 at 11:18 | comment | added | James Martin | Thanks Ryan, and thanks Fedor and Alekk too. I guess Fedor's suggestion of isoperimetric bounds is rather in the same direction as the question of influences. | |
| Jun 2, 2010 at 4:05 | history | answered | Ryan O'Donnell | CC BY-SA 2.5 |