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3. Log-concavity

    1.     A function $f$ is log-concave if $$f(a)f(b) \leq f\left(\frac{a+b}{2}\right)^2.$$ One example of such function is the Gaussian density function. Note that the variables $a,b$ might be vectors or matrices. Log-concavity is fundamental in probability theory and combinatorics, e.g., [MR2213177], [MR4419063].

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      Problem 3.1.

      Consider continuous-time TASEP with step initial condition. There is an associated transition probability function $f(a)$ into the configuration $a$. A question arises whether $f(a)$ is log-concave. While log-concavity appears in many probabilistic models, it remains unclear how to establish it in this context. The motivation includes connections to symmetric functions.


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          Problem 3.2.

          The density of the Tracy–Widom random variable is itself log-concave [MR3721641]. For a large permutation of size $n$, the longest increasing subsequence length $L_n$ often (after suitable centering and scaling) converges to the Tracy–Widom distribution. Does Tracy–Widom log-concavity extend to the finite, discrete models that approximate or ‘discretize’ the Tracy–Widom distribution? Such discrete distributions count permutations by fixed longest increasing subsequence length and may exhibit log-concavity under certain conditions. For a recent development, see [ArXiv:2412.15116].

              Cite this as: AimPL: All roads to the KPZ universality class, available at http://aimpl.org/roadtokpz.