Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study random subgraphs of the n-cube {0,1}n, where nearest-neighbor edges are occupied with probability p. Let pc(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2n/3, where λ is a small positive constant. Let e=n(p−pc(n)). In two previous papers,  shod that the largest component inside a scaling window given by |e|=Θ(2−n/3) is of size Θ(22n/3), below this scaling window it is at most 2(log 2)ne−2, and above this scaling window it is at most O(e2n). ,  prove that for $$p - p_{c} {\left( n \right)} \geqslant e^{{cn^{{1/3}} }}$$ the size of the largest component is at least Θ(e2n), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.
['Christian Borgs', 'Jennifer T. Chayes', 'Remco van der Hofstad', 'Gordon Slade', 'Joel Spencer']
I am writing a research paper on a idea Random Subgraphs Of Finite Graphs: III. I am writing a research paper on a idea The Phase Transition For The n -Cube
[{"from": "human", "value": "I am writing a research paper on a idea Random Subgraphs Of Finite Graphs: III. I am writing a research paper on a idea The Phase Transition For The n -Cube"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study random subgraphs of the n-cube {0,1}n, where nearest-neighbor edges are occupied with probability p. Let pc(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value \u03bb2n/3, where \u03bb is a small positive constant. Let e=n(p\u2212pc(n)). In two previous papers,  shod that the largest component inside a scaling window given by |e|=\u0398(2\u2212n/3) is of size \u0398(22n/3), below this scaling window it is at most 2(log 2)ne\u22122, and above this scaling window it is at most O(e2n). ,  prove that for $$p - p_{c} {\\left( n \\right)} \\geqslant e^{{cn^{{1/3}} }}$$ the size of the largest component is at least \u0398(e2n), which is of the same order as the upper bound. The proof is based on a method that has come to be known as \u201csprinkling,\u201d and relies heavily on the specific geometry of the n-cube."}]