The geometry of polymers and other results in the KPZ universality class

This thesis investigates the geometry of polymers and other miscellaneous results in the Kardar-Parisi-Zhang (KPZ) universality class. Directed polymers have enjoyed a rich history in both probability theory and mathematical physics and have connections to several families of statistical mechanical and random growth models that belong to the KPZ universality class [77]. In this thesis, we focus on 2 integrable polymer models, the (1+1)-dimensional continuum directed random polymer (CDRP) and the half-space log-gamma (HSLG) polymer, and study their path properties. For the CDRP, we show both of its superdiffusivity and localization features. Namely, the annealed law of polymer of length t, upon t²/³ superdiffusive scaling, is tight in the space of C( [0, 1])-valued random variables and the quenched law of any point distance pt from the origin on the path a point-to-point polymer (or the endpoint of a point-to-line polymer) concentrates in a O(1) window around a random favorite point Mp,t. The former marks the first pathwise tightness result for positive temperature models and the latter result confirms the “favorite region conjecture” for the CDRP. Moreover, we provide an explicit random density for the quenched distribution around the favorite point Mp,t. The proofs of both results utilize connections with the KPZ equation and our techniques also allow us to prove properties of the KPZ equation itself, such as ergodicity and limiting Bessel behaviors around the maximum. For the HSLG polymers, we combine our localization techniques from the CDRP and the recently developed HSLG line ensemble results [22, 27] with an innovative combinatorial argument to obtain its limiting quenched endpoint distribution from the diagonal in the boundphase (α < 0). This result proves Kardar’s “pinning” conjecture in the case of HSLG polymers[158]. Finally, this thesis also contains two separate works on the tightness of the Bernoulli Gibbsian line ensemble under mild conditions and the upper-tail large deviation principle (LDP) of the asymmetric simple exclusion process (ASEP) with step initial data. In the first work, we prove that under a mild but uniform control of the one-point marginals of the top curve of the line ensemble, i.e. the shape of the top curve as approximately an inverse parabola and asymptotically covering the entire real line after scaling and recentering, the sequence of line ensembles is tight. With a characterization of [109], our tightness result implies the convergence of the Bernoulli Gibbsian line ensemble to the parabolic Airy line ensemble if the top curve converges to the parabolic Airy2 process in the finite dimensional sense. Compared to a similar work of [93], our result applies to line ensembles with possibly random initial and terminal data, instead of a packed initial condition, and does not rely on exact formulas. In our work on the ASEP, we obtain the exact Lyapunov exponent for the height function of ASEP with step initial data and subsequently its upper-tail LDP, where the rate function matches with that of the TASEP given in a variational form in [156]

A Study of the Interaction between Cucurbit[7]uril and Alkyl Substituted 4-Pyrrolidinopyridinium Salts

The interaction between cucurbit[7]uril (Q[7]) and a series of 4-pyrrolidinopyridinium salts bearing aliphatic substituents at the pyridinium nitrogen, namely 4-(C4H8N)C5H5NRBr, where R = H (C0), Et (C2), n-butyl (C4), n-hexyl (C6), has been studied in aqueous solution by 1H NMR spectroscopy, electronic absorption spectroscopy, and mass spectrometry

The Log Convex Density Conjecture in Hyperbolic Space

The Euclidean log convex density theorem, proved by Gregory Chambers in 2015, asserts that in Euclidean space with a log convex density spheres about the origin are isoperimetric. We provide a partial extension to hyperbolic space in which volume and perimeter densities are related but different

The Log Convex Density Conjecture in Hyperbolic Space

The Euclidean log convex density theorem, proved by Gregory Chambers in 2015, asserts that in Euclidean space with a log convex density spheres about the origin are isoperimetric. We provide a partial extension to hyperbolic space in which volume and perimeter densities are related but different

Amino acid recognition by a fluorescent chemosensor based on cucurbit[8]uril and acridine hydrochloride

A new fluorescent chemosensor comprised of cucurbit[8]uril (Q[8]) and acridine hydrochloride (AC) has been designed and utilized for the recognition of amino acids. The AC was encapsulated by the Q[8] cavity and formed a 1:2 host-guest inclusion complex both in solution (aqueous) and in the solid-state. Whilst free AC is known to be strongly fluorescent, this strong fluorescence was quenched in the inclusion complex Q[8]-AC. This non-fluorescent complex Q[8]-AC was capable of serving as a fluorescence “off-on” probe, and was able to recognize either L-Phe or L-Trp via the competitive interaction between L-Phe or L-Trp. Moreover, the pH responsive nature of the probe allowed for the detection of basic amino acids, namely L-Arg, L-His, or L-Lys). As a result, a fluorescence method for the detection of five amino acids using a single system has been developed

A hemicyanine and cucurbit[n]uril inclusion complex: competitive guest binding of cucurbit[7]uril and cucurbit[8]uril

The interaction between the hemicyanine indole derivative H and the cucubit[n]urils Q[7] and Q[8] has been studied using 1H NMR and UV spectroscopy as well as by fluorescence experiments. Competitive studies on the inclusion of H by Q[7] and Q[8] have also been conducted, and reveal that on changing the size of the Q[n] cavity, the binding behaviour can be very different