We record a conditional form of the $ \epsilon $-dependent Park–Pham theorem. If a monotone property $ \mathcal{F}\subseteq 2^X $ is positively correlated with a conditioning event $ B\subseteq 2^X $ under the product measure $ \mu_p $, then the usual Park–Pham lower bound for $ \mu_p(\mathcal{F}) $ transfers to the conditional probability $ \mathbb P(X_p\in\mathcal{F}\mid X_p\in B) $. This gives, in particular, conditional Park–Pham bounds for increasing conditioning events by Harris's inequality, and for nonmonotone conditioning events that are independent of the target property. We also formulate the transfer principle for finite posets embedded in Boolean lattices and illustrate it with pattern-containment upper sets in permutation classes. Finally, we prove that, for every finite poset $ P $ and every nonempty proper upper set $ U\subseteq\mathcal{I}(P) $, the lattice of order ideals $ \mathcal{I}(P) $ is positively correlated under $ \mu_p $ with the Boolean upper set $ \langle U\rangle\subseteq 2^P $ generated by $ U $. Via Birkhoff's representation theorem, this yields a Park–Pham bound on every finite distributive lattice for the natural one-parameter probability measure induced by conditioning $ \mu_p $ on its canonical order-ideal representation.
Citation: Bryce Alan Christopherson, Darian Colgrove. Conditional Park–Pham bounds under positive correlation[J]. AIMS Mathematics, 2026, 11(11): 36119-36133. doi: 10.3934/math.20261399
We record a conditional form of the $ \epsilon $-dependent Park–Pham theorem. If a monotone property $ \mathcal{F}\subseteq 2^X $ is positively correlated with a conditioning event $ B\subseteq 2^X $ under the product measure $ \mu_p $, then the usual Park–Pham lower bound for $ \mu_p(\mathcal{F}) $ transfers to the conditional probability $ \mathbb P(X_p\in\mathcal{F}\mid X_p\in B) $. This gives, in particular, conditional Park–Pham bounds for increasing conditioning events by Harris's inequality, and for nonmonotone conditioning events that are independent of the target property. We also formulate the transfer principle for finite posets embedded in Boolean lattices and illustrate it with pattern-containment upper sets in permutation classes. Finally, we prove that, for every finite poset $ P $ and every nonempty proper upper set $ U\subseteq\mathcal{I}(P) $, the lattice of order ideals $ \mathcal{I}(P) $ is positively correlated under $ \mu_p $ with the Boolean upper set $ \langle U\rangle\subseteq 2^P $ generated by $ U $. Via Birkhoff's representation theorem, this yields a Park–Pham bound on every finite distributive lattice for the natural one-parameter probability measure induced by conditioning $ \mu_p $ on its canonical order-ideal representation.
| [1] |
J. Park, Threshold phenomena for random discrete structures, Not. Am. Math. Soc., 70 (2023), 1615–1625. https://doi.org/10.1090/noti2802 doi: 10.1090/noti2802
|
| [2] |
J. Park, H. Pham, A proof of the Kahn-Kalai conjecture, J. Am. Math. Soc., 37 (2024), 235–243. https://doi.org/10.1090/jams/1028 doi: 10.1090/jams/1028
|
| [3] |
P. Tran, V. Vu, A short proof of Kahn-Kalai conjecture, Electron. J. Comb., 31 (2024), 3.2. https://doi.org/10.37236/12266 doi: 10.37236/12266
|
| [4] |
J. Kahn, G. Kalai, Thresholds and expectation thresholds, Comb. Probab. Comput., 16 (2007), 495–502. https://doi.org/10.1017/S0963548307008474 doi: 10.1017/S0963548307008474
|
| [5] |
T. Bell, The Park–Pham theorem with optimal convergence rate, Electron. J. Comb., 30 (2023), 2.25. https://doi.org/10.37236/11600 doi: 10.37236/11600
|
| [6] |
K. Frankston, J. Kahn, B. Narayanan, J. Park, Thresholds versus fractional expectation-thresholds, Ann. Math., 194 (2021), 475–495. https://doi.org/10.4007/annals.2021.194.2.2 doi: 10.4007/annals.2021.194.2.2
|
| [7] |
B. Park, J. Vondrák, A simple proof of the nonuniform Kahn–Kalai conjecture, SIAM J. Discrete Math., 38 (2024), 2089–2094. https://doi.org/10.1137/23M1587075 doi: 10.1137/23M1587075
|
| [8] |
T. Przybyłowski, O. Riordan, Thresholds, expectation thresholds and cloning, Electron. J. Comb., 31 (2024), 4.74. https://doi.org/10.37236/12825 doi: 10.37236/12825
|
| [9] |
B. T. Graham, G. R. Grimmett, Influence and sharp-threshold theorems for monotonic measures, Ann. Probab., 34 (2006), 1726–1745. https://doi.org/10.1214/009117906000000278 doi: 10.1214/009117906000000278
|
| [10] |
B. Gunby, X. He, B. Narayanan, Down-set thresholds, Random Struct. Algorithms, 63 (2023), 442–456. https://doi.org/10.1002/rsa.21148 doi: 10.1002/rsa.21148
|
| [11] |
T. Harris, A lower bound for the critical probability in a certain percolation process, Math. Proc. Camb. Philos. Soc., 56 (1960), 13–20. https://doi.org/10.1017/S0305004100034241 doi: 10.1017/S0305004100034241
|
| [12] |
J. D. Esary, F. Proschan, D. W. Walkup, Association of random variables, with applications, Ann. Math. Stat., 38 (1967), 1466–1474. https://doi.org/10.1214/aoms/1177698701 doi: 10.1214/aoms/1177698701
|
| [13] |
C. M. Fortuin, P. W. Kasteleyn, J. Ginibre, Correlation inequalities on some partially ordered sets, Commun. Math. Phys., 22 (1971), 89–103. https://doi.org/10.1007/BF01651330 doi: 10.1007/BF01651330
|
| [14] |
G. Birkhoff, Rings of sets, Duke Math. J., 3 (1937), 443–454. https://doi.org/10.1215/S0012-7094-37-00334-X doi: 10.1215/S0012-7094-37-00334-X
|
| [15] |
N. Beerenwinkel, N. Eriksson, B. Sturmfels, Conjunctive Bayesian networks, Bernoulli, 13 (2007), 893–909. https://doi.org/10.3150/07-BEJ6133 doi: 10.3150/07-BEJ6133
|
| [16] |
D. E. Knuth, The art of computer programming, Vol. 1: Fundamental algorithms, Addison-Wesley Professional, 1968. |
| [17] |
B. A. Christopherson, J. Baretz, When do the Kahn–Kalai bounds provide nontrivial information? J. Inequal. Appl., 2025 (2025), 23. https://doi.org/10.1186/s13660-025-03272-z |
Bryce Alan Christopherson, Darian Colgrove. Conditional Park–Pham bounds under positive correlation[J]. AIMS Mathematics, 2026, 11(11): 36119-36133. doi: 10.3934/math.20261399
DownLoad: